Publications

We develop coarse-grained, distance- and orientation-dependent statistical potentials from the growing protein structural databases. For protein structural classes (alpha, beta, and alpha/beta), a substantial number of backbone-backbone and backbone-side-chain contacts stabilize the native folds. By taking into account the importance of backbone interactions with a virtual backbone interaction center as the 21st anisotropic site, we construct a 21 x 21 interaction scheme. The new potentials are studied using spherical harmonics analysis (SHA) and a smooth, continuous version is constructed using spherical harmonic synthesis (SHS). Our approach has the following advantages: (1) The smooth, continuous form of the resulting potentials is more realistic and presents significant advantages for computational simulations, and (2) with SHS, the potential values can be computed efficiently for arbitrary coordinates, requiring only the knowledge of a few spherical harmonic coefficients. The performance of the new orientation-dependent potentials was tested using a standard database of decoy structures. The results show that the ability of the new orientation-dependent potentials to recognize native protein folds from a set of decoy structures is strongly enhanced by the inclusion of anisotropic backbone interaction centers. The anisotropic potentials can be used to develop realistic coarse-grained simulations of proteins, with direct applications to protein design, folding, and aggregation.

Signal transmission at the molecular level in many biological complexes occurs through allosteric transitions. Allostery describes the responses of a complex to binding of ligands at sites that are spatially well separated from the binding region. We describe the structural perturbation method, based on phonon propagation in solids, which can be used to determine the signal-transmitting allostery wiring diagram (AWD) in large but finite-sized biological complexes. Application to the bacterial chaperonin GroEL–GroES complex shows that the AWD determined from structures also drives the allosteric transitions dynamically. From both a structural and dynamical perspective these transitions are largely determined by formation and rupture of salt-bridges. The molecular description of allostery in GroEL provides insights into its function, which is quantitatively described by the iterative annealing mechanism. Remarkably, in this complex molecular machine, a deep connection is established between the structures, reaction cycle during which GroEL undergoes a sequence of allosteric transitions, and function, in a self-consistent manner.

This article is part of a discussion meeting issue ‘Allostery and molecular machines’.

Myosin VI (MVI) is the only known member of the myosin superfamily that, upon dimerization, walks processively toward the pointed end of the actin filament. The leading head of the dimer directs the trailing head forward with a power stroke, a conformational change of the motor domain exaggerated by the lever arm. Using a unique coarse-grained model for the power stroke of a single MVI, we provide the molecular basis for its motility. We show that the power stroke occurs in two major steps. First, the motor domain attains the poststroke conformation without directing the lever arm forward; and second, the lever arm reaches the poststroke orientation by undergoing a rotational diffusion. From the analysis of the trajectories, we discover that the potential that directs the rotating lever arm toward the poststroke conformation is almost flat, implying that the lever arm rotation is mostly uncoupled from the motor domain. Because a backward load comparable to the largest interhead tension in a MVI dimer prevents the rotation of the lever arm, our model suggests that the leading-head lever arm of a MVI dimer is uncoupled, in accord with the inference drawn from polarized total internal reflection fluorescence (polTIRF) experiments. Without any adjustable parameter, our simulations lead to quantitative agreement with polTIRF experiments, which validates the structural insights. Finally, in addition to making testable predictions, we also discuss the implications of our model in explaining the broad step-size distribution of the MVI stepping pattern.

Signaling in enzymatic networks is typically triggered by environmental fluctuations, resulting in a series of stochastic chemical reactions, leading to corruption of the signal by noise. For example, information flow is initiated by binding of extracellular ligands to receptors, which is transmitted through a cascade involving kinase-phosphatase stochastic chemical reactions. For a class of such networks, we develop a general field-theoretic approach to calculate the error in signal transmission as a function of an appropriate control variable. Application of the theory to a simple push-pull network, a module in the kinase-phosphatase cascade, recovers the exact results for error in signal transmission previously obtained using umbral calculus [Hinczewski and Thirumalai, Phys. Rev. X 4, 041017 (2014)]. We illustrate the generality of the theory by studying the minimal errors in noise reduction in a reaction cascade with two connected push-pull modules. Such a cascade behaves as an effective three-species network with a pseudointermediate. In this case, optimal information transfer, resulting in the smallest square of the error between the input and output, occurs with a time delay, which is given by the inverse of the decay rate of the pseudointermediate. Surprisingly, in these examples the minimum error computed using simulations that take nonlinearities and discrete nature of molecules into account coincides with the predictions of a linear theory. In contrast, there are substantial deviations between simulations and predictions of the linear theory in error in signal propagation in an enzymatic push-pull network for a certain range of parameters. Inclusion of second-order perturbative corrections shows that differences between simulations and theoretical predictions are minimized. Our study establishes that a field theoretic formulation of stochastic biological signaling offers a systematic way to understand error propagation in networks of arbitrary complexity.

Folded states of single domain globular proteins are compact with high packing density. The radius of gyration, R_g, of both the folded and unfolded states increase as N^ν where N is the number of amino acids in the protein. The values of the Flory exponent ν are, respectively, ≈⅓ and ≈0.6 in the folded and unfolded states, coinciding with those for homopolymers. However, the extent of compaction of the unfolded state of a protein under low denaturant concentration (collapsibility), conditions favoring the formation of the folded state, is unknown. We develop a theory that uses the contact map of proteins as input to quantitatively assess collapsibility of proteins. Although collapsibility is universal, the propensity to be compact depends on the protein architecture. Application of the theory to over two thousand proteins shows that collapsibility depends not only on N but also on the contact map reflecting the native structure. A major prediction of the theory is that β-sheet proteins are far more collapsible than structures dominated by α-helices. The theory and the accompanying simulations, validating the theoretical predictions, provide insights into the differing conclusions reached using different experimental probes assessing the extent of compaction of proteins. By calculating the criterion for collapsibility as a function of protein length we provide quantitative insights into the reasons why single domain proteins are small and the physical reasons for the origin of multi-domain proteins. Collapsibility of non-coding RNA molecules is similar β-sheet proteins structures adding support to “Compactness Selection Hypothesis”.

Fluctuations in the physical properties of biological machines are inextricably linked to their functions. Distributions of run lengths and velocities of processive molecular motors, like kinesin-1, are accessible through single-molecule techniques, but rigorous theoretical models for these probabilities are lacking. Here, we derive exact analytic results for a kinetic model to predict the resistive force (F)-dependent velocity [P(v)] and run length [P(n)] distribution functions of generic finitely processive molecular motors. Our theory quantitatively explains the zero force kinesin-1 data for both P(n) and P(v) using the detachment rate as the only parameter. In addition, we predict the F dependence of these quantities. At nonzero F, P(v) is non-Gaussian and is bimodal with peaks at positive and negative values of v, which is due to the discrete step size of kinesin-1. Although the predictions are based on analyses of kinesin-1 data, our results are general and should hold for any processive motor, which walks on a track by taking discrete steps.

One of the most intriguing results of single-molecule experiments on proteins and nucleic acids is the discovery of functional heterogeneity: the observation that complex cellular machines exhibit multiple, biologically active conformations. The structural differences between these conformations may be subtle, but each distinct state can be remarkably long-lived, with interconversions between states occurring only at macroscopic timescales, fractions of a second or longer. Although we now have proof of functional heterogeneity in a handful of systems-enzymes, motors, adhesion complexes-identifying and measuring it remains a formidable challenge. Here, we show that evidence of this phenomenon is more widespread than previously known, encoded in data collected from some of the most well-established single-molecule techniques: atomic force microscopy or optical tweezer pulling experiments. We present a theoretical procedure for analyzing distributions of rupture/unfolding forces recorded at different pulling speeds. This results in a single parameter, quantifying the degree of heterogeneity, and also leads to bounds on the equilibration and conformational interconversion timescales. Surveying 10 published datasets, we find heterogeneity in 5 of them, all with interconversion rates slower than 10 s(-1) Moreover, we identify two systems where additional data at realizable pulling velocities is likely to find a theoretically predicted, but so far unobserved crossover regime between heterogeneous and nonheterogeneous behavior. The significance of this regime is that it will allow far more precise estimates of the slow conformational switching times, one of the least understood aspects of functional heterogeneity.

Although it is known that single-domain proteins fold and unfold by parallel pathways, demonstration of this expectation has been difficult to establish in experiments. Unfolding rate, [Formula: see text], as a function of force f, obtained in single-molecule pulling experiments on src SH3 domain, exhibits upward curvature on a [Formula: see text] plot. Similar observations were reported for other proteins for the unfolding rate [Formula: see text]. These findings imply unfolding in these single-domain proteins involves a switch in the pathway as f or [Formula: see text] is increased from a low to a high value. We provide a unified theory demonstrating that if [Formula: see text] as a function of a perturbation (f or [Formula: see text]) exhibits upward curvature then the underlying energy landscape must be strongly multidimensional. Using molecular simulations we provide a structural basis for the switch in the pathways and dramatic shifts in the transition-state ensemble (TSE) in src SH3 domain as f is increased. We show that a single-point mutation shifts the upward curvature in [Formula: see text] to a lower force, thus establishing the malleability of the underlying folding landscape. Our theory, applicable to any perturbation that affects the free energy of the protein linearly, readily explains movement in the TSE in a β-sandwich (I27) protein and single-chain monellin as the denaturant concentration is varied. We predict that in the force range accessible in laser optical tweezer experiments there should be a switch in the unfolding pathways in I27 or its mutants.

Conventional kinesin walks by a hand-over-hand mechanism on the microtubule (MT) by taking ∼8 nm discrete steps and consumes one ATP molecule per step. The time needed to complete a single step is on the order of 20 μs. We show, using simulations of a coarse-grained model of the complex containing the two motor heads, the MT and the coiled coil, that to obtain quantitative agreement with experiments for the stepping kinetics hydrodynamic interactions (HIs) have to be included. In simulations without hydrodynamic interactions, spanning nearly 20 μs, not a single step was completed in one hundred trajectories. In sharp contrast, nearly 14% of the steps reached the target binding site within 6 μs when HIs were included. Somewhat surprisingly, there are qualitative differences in the diffusion pathways in simulations with and without HI. The extent of movement of the trailing head of kinesin on the MT during the diffusion stage of stepping is considerably greater in simulations with HI than in those without HI. It is likely that inclusion of HI is crucial in the accurate description of motility of other motors as well.

Genes and proteins regulate cellular functions through complex circuits of biochemical reactions. Fluctuations in the components of these regulatory networks result in noise that invariably corrupts the signal, possibly compromising function. Here, we create a practical formalism based on ideas introduced by Wiener and Kolmogorov (WK) for filtering noise in engineered communications systems to quantitatively assess the extent to which noise can be controlled in biological processes involving negative feedback. Application of the theory, which reproduces the previously proven scaling of the lower bound for noise suppression in terms of the number of signaling events, shows that a tetracycline repressor-based negative-regulatory gene circuit behaves as a WK filter. For the class of Hill-like nonlinear regulatory functions, this type of filter provides the optimal reduction in noise. Our theoretical approach can be readily combined with experimental measurements of response functions in a wide variety of genetic circuits, to elucidate the general principles by which biological networks minimize noise.

Lifetimes of bound states of protein complexes or biomolecule folded states typically decrease when subject to mechanical force. However, a plethora of biological systems exhibit the counter-intuitive phenomenon of catch bonding, where non-covalent bonds become stronger under externally applied forces. The quest to understand the origin of catch-bond behavior has led to the development of phenomenological and microscopic theories that can quantitatively recapitulate experimental data. Here, we assess the successes and limitations of such theories in explaining experimental data. The most widely applied approach is a phenomenological two-state model, which fits all of the available data on a variety of complexes: actomyosin, kinetochore-microtubule, selectin-ligand, and cadherin-catenin binding to filamentous actin. With a primary focus on the selectin family of cell-adhesion complexes, we discuss the positives and negatives of phenomenological models and the importance of evaluating the physical relevance of fitting parameters. We describe a microscopic theory for selectins, which provides a structural basis for catch bonds and predicts a crucial allosteric role for residues Asn82-Glu88. We emphasize the need for new theories and simulations that can mimic experimental conditions, given the complex response of cell adhesion complexes to force and their potential role in a variety of biological contexts.

Because of the potential link between -1 programmed ribosomal frameshifting and response of a pseudoknot (PK) RNA to force, a number of single-molecule pulling experiments have been performed on PKs to decipher the mechanism of programmed ribosomal frameshifting. Motivated in part by these experiments, we performed simulations using a coarse-grained model of RNA to describe the response of a PK over a range of mechanical forces (fs) and monovalent salt concentrations (Cs). The coarse-grained simulations quantitatively reproduce the multistep thermal melting observed in experiments, thus validating our model. The free energy changes obtained in simulations are in excellent agreement with experiments. By varying f and C, we calculated the phase diagram that shows a sequence of structural transitions, populating distinct intermediate states. As f and C are changed, the stem-loop tertiary interactions rupture first, followed by unfolding of the 3′-end hairpin (I⇌F). Finally, the 5′-end hairpin unravels, producing an extended state (E⇌I). A theoretical analysis of the phase boundaries shows that the critical force for rupture scales as (logCm)(α) with α=1(0.5) for E⇌I (I⇌F) transition. This relation is used to obtain the preferential ion-RNA interaction coefficient, which can be quantitatively measured in single-molecule experiments, as done previously for DNA hairpins. A by-product of our work is the suggestion that the frameshift efficiency is likely determined by the stability of the 5′-end hairpin that the ribosome first encounters during translation.

Recent experiments showing scaling of the intrachromosomal contact probability, P(s)∼s(-1) with the genomic distance s, are interpreted to mean a self-similar fractal-like chromosome organization. However, scaling of P(s) varies across organisms, requiring an explanation. We illustrate dynamical arrest in a highly confined space as a discriminating marker for genome organization, by modeling chromosomes inside a nucleus as a homopolymer confined to a sphere of varying sizes. Brownian dynamics simulations show that the chain dynamics slows down as the polymer volume fraction (ϕ) inside the confinement approaches a critical value ϕ(c). The universal value of ϕ(c)(∞)≈0.44 for a sufficiently long polymer (N≫1) allows us to discuss genome dynamics using ϕ as the sole parameter. Our study shows that the onset of glassy dynamics is the reason for the segregated chromosome organization in humans (N≈3×10(9), ϕ≳ϕ(c)(∞)), whereas chromosomes of budding yeast (N≈10(8), ϕ<ϕ(c)(∞)) are equilibrated with no clear signature of such organization.

Folding of Ubiquitin (Ub), a functionally important protein found in eukaryotic organisms, is investigated at low and neutral pH at different temperatures using simulations of the coarse-grained self-organized-polymer model with side chains (SOP-SC). The melting temperatures (Tm’s), identified with the peaks in the heat capacity curves, decrease as pH decreases, in qualitative agreement with experiments. The calculated radius of gyration, showing dramatic variations with pH, is in excellent agreement with scattering experiments. At Tm, Ub folds in a two-state manner at low and neutral pH. Clustering analysis of the conformations sampled in equilibrium folding trajectories at Tm, with multiple transitions between the folded and unfolded states, shows a network of metastable states connecting the native and unfolded states. At low and neutral pH, Ub folds with high probability through a preferred set of conformations resulting in a pH-dependent dominant folding pathway. Folding kinetics reveal that Ub assembly at low pH occurs by multiple pathways involving a combination of nucleation-collapse and diffusion collision mechanism. The mechanism by which Ub folds is dictated by the stability of the key secondary structural elements responsible for establishing long-range contacts and collapse of Ub. Nucleation collapse mechanism holds if the stability of these elements are marginal, as would be the case at elevated temperatures. If the lifetimes associated with these structured microdomains are on the order of hundreds of microseconds, then Ub folding follows the diffusion-collision mechanism with intermediates, many of which coincide with those found in equilibrium. Folding at neutral pH is a sequential process with a populated intermediate resembling that sampled at equilibrium. The transition state structures, obtained using a Pfold analysis, are homogeneous and globular with most of the secondary and tertiary structures being native-like. Many of our findings for both the thermodynamics and kinetics of folding are not only in agreement with experiments but also provide missing details not resolvable in standard experiments. The key prediction that folding mechanism varies dramatically with pH is amenable to experimental tests.

Helicases, involved in a number of cellular functions, are motors that translocate along single-stranded nucleic acid and couple the motion to unwinding double-strands of a duplex nucleic acid. The junction between double- and single-strands creates a barrier to the movement of the helicase, which can be manipulated in vitro by applying mechanical forces directly on the nucleic acid strands. Single-molecule experiments have demonstrated that the unwinding velocities of some helicases increase dramatically with increase in the external force, while others show little response. In contrast, the unwinding processivity always increases when the force increases. The differing responses of the unwinding velocity and processivity to force have lacked explanation. By generalizing a previous model of processive unwinding by helicases, we provide a unified framework for understanding the dependence of velocity and processivity on force and the nucleic acid sequence. We predict that the sensitivity of unwinding processivity to external force is a universal feature that should be observed in all helicases. Our prediction is illustrated using T7 and NS3 helicases as case studies. Interestingly, the increase in unwinding processivity with force depends on whether the helicase forces basepair opening by direct interaction or if such a disruption occurs spontaneously due to thermal fluctuations. Based on the theoretical results, we propose that proteins like single-strand binding proteins associated with helicases in the replisome may have coevolved with helicases to increase the unwinding processivity even if the velocity remains unaffected.

We investigate the conformations of DNA-like stiff chains, characterized by contour length (L) and persistence length (lp), in a variety of crowded environments containing monodisperse soft spherical (SS) and spherocylindrical (SC) particles, a mixture of SS and SC, and a milieu mimicking the composition of proteins in the Escherichia coli cytoplasm. The stiff chain, whose size modestly increases in SS crowders up to ϕ ≈ 0.1, is considerably more compact at low volume fractions (ϕ ≤ 0.2) in monodisperse SC particles than in a medium containing SS particles. A 1:1 mixture of SS and SC crowders induces greater chain compaction than the pure SS or SC crowders at the same ϕ, with the effect being highly nonadditive. We also discover a counterintuitive result that the polydisperse crowding environment, mimicking the composition of a cell lysate, swells the DNA-like polymer, which is in stark contrast to the size reduction of flexible polymers in the same milieu. Trapping of the stiff chain in a fluctuating tube-like environment created by large-sized crowders explains the dramatic increase in size and persistence length of the stiff chain. In the polydisperse medium, mimicking the cellular environment, the size of the DNA (or related RNA) is determined by L/lp. At low L/lp, the size of the polymer is unaffected, whereas there is a dramatic swelling at an intermediate value of L/lp. We use these results to provide insights into recent experiments on crowding effects on RNA and also make testable predictions.

Riboswitches, RNA elements found in the untranslated region, regulate gene expression by binding to target metaboloites with exquisite specificity. Binding of metabolites to the conserved aptamer domain allosterically alters the conformation in the downstream expression platform. The fate of gene expression is determined by the changes in the downstream RNA sequence. As the metabolite-dependent cotranscriptional folding and unfolding dynamics of riboswitches are the key determinant of gene expression, it is important to investigate both the thermodynamics and kinetics of riboswitches both in the presence and absence of metabolite. Single molecule force experiments that decipher the free energy landscape of riboswitches from their mechanical responses, theoretical and computational studies have recently shed light on the distinct mechanism of folding dynamics in different classes of riboswitches. Here, we first discuss the dynamics of water around riboswitch, highlighting that water dynamics can enhance the fluctuation of nucleic acid structure. To go beyond native state fluctuations, we used the Self-Organized Polymer model to predict the dynamics of add adenine riboswitch under mechanical forces. In addition to quantitatively predicting the folding landscape of add-riboswitch, our simulations also explain the difference in the dynamics between pbuE adenine- and add adenine-riboswitches. In order to probe the function in vivo, we use the folding landscape to propose a system level kinetic network model to quantitatively predict how gene expression is regulated for riboswitches that are under kinetic control.

Experiments show that macromolecular crowding modestly reduces the size of intrinsically disordered proteins even at a volume fraction (ϕ) similar to that in the cytosol, whereas DNA undergoes a coil-to-globule transition at very small ϕ. We show using a combination of scaling arguments and simulations that the polymer size R̅(g)(ϕ) depends on x=R̅(g)(0)/D, where D is the ϕ-dependent distance between the crowders. If x≲O(1), there is only a small decrease in R̅(g)(ϕ) as ϕ increases. When x≫O(1), a cooperative coil-to-globule transition is induced. Our theory quantitatively explains a number of experiments.

The Anfinsen principle that the protein sequence uniquely determines its structure is based on experiments on oxidative refolding of a protein with disulfide bonds. The problem of how protein folding drives disulfide bond formation is poorly understood. Here, we have solved this long-standing problem by creating a general method for implementing the chemistry of disulfide bond formation and rupture in coarse-grained molecular simulations. As a case study, we investigate the oxidative folding of bovine pancreatic trypsin inhibitor (BPTI). After confirming the experimental findings that the multiple routes to the folded state contain a network of states dominated by native disulfides, we show that the entropically unfavorable native single disulfide [14-38] between Cys14 and Cys38 forms only after polypeptide chain collapse and complete structuring of the central core of the protein containing an antiparallel β-sheet. Subsequent assembly, resulting in native two-disulfide bonds and the folded state, involves substantial unfolding of the protein and transient population of nonnative structures. The rate of [14-38] formation increases as the β-sheet stability increases. The flux to the native state, through a network of kinetically connected native-like intermediates, changes dramatically by altering the redox conditions. Disulfide bond formation between Cys residues not present in the native state are relevant only on the time scale of collapse of BPTI. The finding that formation of specific collapsed native-like structures guides efficient folding is applicable to a broad class of single-domain proteins, including enzyme-catalyzed disulfide proteins.

Ribozymes, which carry out phosphoryl-transfer reactions, often require Mg²⁺ ions for catalytic activity. The correct folding of the active site and ribozyme tertiary structure is also regulated by metal ions in a manner that is not fully understood. Here we employ coarse-grained molecular simulations to show that individual structural elements of the group I ribozyme from the bacterium Azoarcus form spontaneously in the unfolded ribozyme even at very low Mg²⁺ concentrations, and are transiently stabilized by the coordination of Mg²⁺ ions to specific nucleotides. However, competition for scarce Mg²⁺ and topological constraints that arise from chain connectivity prevent the complete folding of the ribozyme. A much higher Mg²⁺ concentration is required for complete folding of the ribozyme and stabilization of the active site. When Mg²⁺ is replaced by Ca²⁺ the ribozyme folds, but the active site remains unstable. Our results suggest that group I ribozymes utilize the same interactions with specific metal ligands for both structural stability and chemical activity.

Expression of a large fraction of genes in bacteria is controlled by riboswitches, which are found in the untranslated region of mRNA. Structurally riboswitches have a conserved aptamer domain to which a metabolite binds, resulting in a conformational change in the downstream expression platform. Prediction of the functions of riboswitches requires a quantitative description of the folding landscape so that the barriers and time scales for the conformational change in the switching region in the aptamer can be estimated. Using a combination of all atom molecular dynamics (MD) and coarse-grained model simulations we studied the response of adenine (A) binding add and pbuE A-riboswitches to mechanical force. The two riboswitches contain a structurally similar three-way junction formed by three paired helices, P1, P2, and P3, but carry out different functions. Using pulling simulations, with structures generated in MD simulations, we show that after P1 rips the dominant unfolding pathway in the add A-riboswitch is the rupture of P2 followed by unraveling of P3. In the pbuE A-riboswitch, after P1 unfolds P3 ruptures ahead of P2. The order of unfolding of the helices, which is in accord with single molecule pulling experiments, is determined by the relative stabilities of the individual helices. Our results show that the stability of isolated helices determines the order of assembly and response to force in these non-coding regions. We use the simulated free energy profile for the pbuE A-riboswitch to estimate the time scale for allosteric switching, which shows that this riboswitch is under kinetic control lending additional support to the conclusion based on single molecule pulling experiments. A consequence of the stability hypothesis is that a single point mutation (U28C) in the P2 helix of the add A-riboswitch, which increases the stability of P2, would make the folding landscapes of the two riboswitches similar. This prediction can be tested in single molecule pulling experiments.

The amyloid β (Aβ) peptide associated with Alzheimer’s disease results from processing of the amyloid precursor protein (APP) by secretases. Cleavage of APP by β-secretase produces a 99 amino acid C-terminal fragment of APP (C99) consisting of a single transmembrane (TM) helix. Simulations of C99 congeners and structural studies of C99 in surfactant micelles and lipid vesicles have shown that a key peptide structural motif is a prominent “GG kink,” centered at two glycines dividing the TM helix. The flexibility of the GG kink is important in the processing of C99 by γ-secretase. We performed multiscale simulations of C99(15-55) in a DPC surfactant micelle and POPC lipid bilayer in order to elucidate the role of membrane surface curvature in modulating the peptide structure. C99(15-55) in a DPC surfactant micelle possesses a “GG kink,” in the TM domain near the dynamic hinge located at G37/G38. Such a kink is not observed in C99(15-55) in a POPC lipid bilayer. Intramolecular interaction between the extracellular and TM domains of C99(15-55) is enhanced in the micelle environment, influencing helical stability, TM helix extension, exposure to water, and depth of insertion in the lipophilic region. Our results show that the fluctuations of the structural ensemble of APP are strongly influenced by membrane surface curvature.

Enhanced dynamical fluctuations of RNAs, facilitated by a network of water molecules with strong interactions with RNA, are suspected to be critical in their ability to respond to a variety of cellular signals. Using atomically detailed molecular dynamics simulations at various temperatures of purine (adenine) and preQ1 sensing riboswitch aptamers, which control gene expression by sensing and binding to metabolites, we show that water molecules in the vicinity of RNAs undergo complex dynamics depending on the local structures of the RNAs. The overall lifetimes of hydrogen bonds (HBs) of surface-bound waters are more than at least 1-2 orders of magnitude longer than those of bulk water. Slow hydration dynamics, revealed in the non-Arrhenius behavior of the relaxation time, arises from high activation barriers to break water HBs with a nucleotide and by reduced diffusion of water. The relaxation kinetics at specific locations in the two RNAs show a broad spectrum of time scales reminiscent of glass-like behavior, suggesting that the hydration dynamics is highly heterogeneous. Both RNAs undergo dynamic transition at T = TD ≳ 200 K, as assessed by the mean-square fluctuation of hydrogen atoms ⟨x(2)⟩, which undergoes an abrupt harmonic-to-anharmonic transition at TD. The near-universal value of TD found for these RNAs and previously for tRNA is strongly correlated with changes in hydration dynamics as T is altered. Hierarchical dynamics of waters associated with the RNA surface, revealed in the motions of distinct classes of water with well-separated time scales, reflects the heterogeneous local environment on the molecular surface of RNA. At low temperatures, slow water dynamics predominates over structural transitions. Our study demonstrates that the complex interplay of dynamics between water and the local environment in the RNA structures could be a key determinant of the functional activities of RNA.

Understanding the stabilities of proteins in nanopores requires a quantitative description of confinement induced interactions between amino acid side chains. We use molecular dynamics simulations to study the nature of interactions between the side chain pairs ALA-PHE, SER-ASN, and LYS-GLU in bulk water and in water-filled nanopores. The temperature dependence of the bulk solvent potentials of mean force and the interaction free energies in cylindrical and spherical nanopores is used to identify the corresponding entropic and enthalpic components. The entropically stabilized hydrophobic interaction between ALA and PHE in bulk water is enthalpically dominated upon confinement depending on the relative orientations between the side chains. In the case of SER-ASN, hydrogen bonded configurations that are similar in bulk water are thermodynamically distinct in a cylindrical pore, thus making rotamer distributions different from those in the bulk. Remarkably, salt bridge formation between LYS-GLU is stabilized by entropy in contrast to the bulk. Implications of our findings for confinement-induced alterations in protein stability are briefly outlined.

Heterogeneity in biological molecules, resulting in molecule-to-molecule variations in their dynamics and function, is an emerging theme. To elucidate the consequences of heterogeneous behavior at the single molecule level, we propose an exactly solvable model in which the unfolding rate due to mechanical force depends parametrically on an auxiliary variable representing an entropy barrier arising from fluctuations in internal dynamics. When the rate of fluctuations–a measure of dynamical disorder–is comparable to or smaller than the rate of force-induced unbinding, we show that there are two experimentally observable consequences: nonexponential survival probability at constant force, and a heavy-tailed rupture force distribution at constant loading rate. By fitting our analytical expressions to data from single molecule pulling experiments on proteins and DNA, we quantify the extent of disorder. We show that only by analyzing data over a wide range of forces and loading rates can the role of disorder due to internal dynamics be quantitatively assessed.

Mechanical forces acting on cell adhesion receptor proteins regulate a range of cellular functions by formation and rupture of noncovalent interactions with ligands. Typically, force decreases the lifetimes of intact complexes (“slip bonds”), making the discovery that these lifetimes can also be prolonged (“catch bonds”) a surprise. We created a microscopic analytic theory by incorporating the structures of selectin and integrin receptors into a conceptual framework based on the theory of stochastic equations, which quantitatively explains a wide range of experimental data (including catch bonds at low forces and slip bonds at high forces). Catch bonds arise due to force-induced remodeling of hydrogen bond networks, a finding that also accounts for unbinding in structurally unrelated integrin-fibronectin and actomyosin complexes. For the selectin family, remodeling of hydrogen bond networks drives an allosteric transition resulting in the formation of the maximum number of hydrogen bonds determined only by the structure of the receptor and independent of the ligand. A similar transition allows us to predict the increase in the number of hydrogen bonds in a particular allosteric state of α5β1 integrin-fibronectin complex, a conformation which is yet to be crystallized. We also make a testable prediction that a single point mutation (Tyr51Phe) in the ligand associated with selectin should dramatically alter the nature of the catch bond compared with the wild type. Our work suggests that nature uses a ductile network of hydrogen bonds to engineer function over a broad range of forces.

Protein aggregation, linked to many of diseases, is initiated when monomers access rogue conformations that are poised to form amyloid fibrils. We show, using simulations of src SH3 domain, that mechanical force enhances the population of the aggregation-prone (N(⁎)) states, which are rarely populated under force free native conditions but are encoded in the spectrum of native fluctuations. The folding phase diagrams of SH3 as a function of denaturant concentration ([C]), mechanical force (f), and temperature exhibit an apparent two-state behavior, without revealing the presence of the elusive N(⁎) states. Interestingly, the phase boundaries separating the folded and unfolded states at all [C] and f fall on a master curve, which can be quantitatively described using an analogy to superconductors in a magnetic field. The free energy profiles as a function of the molecular extension (R), which are accessible in pulling experiments, (R), reveal the presence of a native-like N(⁎) with a disordered solvent-exposed amino-terminal β-strand. The structure of the N(⁎) state is identical with that found in Fyn SH3 by NMR dispersion experiments. We show that the timescale for fibril formation can be estimated from the population of the N(⁎) state, determined by the free energy gap separating the native structure and the N(⁎) state, a finding that can be used to assess fibril forming tendencies of proteins. The structures of the N(⁎) state are used to show that oligomer formation and likely route to fibrils occur by a domain-swap mechanism in SH3 domain.

Intermediate filaments (IFs) are key to the mechanical strength of metazoan cells. Their basic building blocks are dimeric coiled coils mediating hierarchical assembly of the full-length filaments. Here we use single-molecule force spectroscopy by optical tweezers to assess the folding and stability of coil 2B of the model IF protein vimentin. The coiled coil was unzipped from its N and C termini. When pulling from the C terminus, we observed that the coiled coil was resistant to force owing to the high stability of the C-terminal region. Pulling from the N terminus revealed that the N-terminal half is considerably less stable. The mechanical pulling assay is a unique tool to study and control seed formation and structure propagation of the coiled coil. We then used rigorous theory-based deconvolution for a model-free extraction of the energy landscape and local stability profiles. The data obtained from the two distinct pulling directions complement each other and reveal a tripartite stability of the coiled coil: a labile N-terminal half, followed by a medium stability section and a highly stable region at the far C-terminal end. The different stability regions provide important insight into the mechanics of IF assembly.

The 99 amino acid C-terminal fragment of amyloid precursor protein (C99), consisting of a single transmembrane (TM) helix, is known to form homodimers. Homodimers can be processed by γ-secretase to produce amyloid-β (Aβ) protein, which is implicated in Alzheimer’s disease (AD). While knowledge of the structure of C99 homodimers is of great importance, experimental NMR studies and simulations have produced varying structural models, including right-handed and left-handed coiled-coils. In order to investigate the structure of this critical protein complex, simulations of the C99(15-55) homodimer in POPC membrane bilayer and DPC surfactant micelle environments were performed using a multiscale approach that blends atomistic and coarse-grained models. The C99(15-55) homodimer adopts a dominant right-handed coiled-coil topology consisting of three characteristic structural states in a bilayer, only one of which is dominant in the micelle. Our structural study, which provides a self-consistent framework for understanding a number of experiments, shows that the energy landscape of the C99 homodimer supports a variety of slowly interconverting structural states. The relative importance of any given state can be modulated through environmental selection realized by altering the membrane or micelle characteristics.

Concepts rooted in physics are becoming increasingly important in biology as we transition to an era in which quantitative descriptions of all processes from molecular to cellular level are needed. In this perspective I discuss two unexpected findings of universal behavior, uncommon in biology, in the self-assembly of proteins and RNA. These findings, which are surprising, reveal that physics ideas applied to biological problems, ranging from folding to gene expression to cellular movement and communication between cells, might lead to discovery of universal principles operating in adoptable living systems.

Cellular signaling involves the transmission of environmental information through cascades of stochastic biochemical reactions, inevitably introducing noise that compromises signal fidelity. Each stage of the cascade often takes the form of a kinase-phosphatase push-pull network, a basic unit of signaling pathways whose malfunction is linked with a host of cancers. We show that this ubiquitous enzymatic network motif effectively behaves as a Wiener-Kolmogorov optimal noise filter. Using concepts from umbral calculus, we generalize the linear Wiener-Kolmogorov theory, originally introduced in the context of communication and control engineering, to take nonlinear signal transduction and discrete molecule populations into account. This allows us to derive rigorous constraints for efficient noise reduction in this biochemical system. Our mathematical formalism yields bounds on filter performance in cases important to cellular function—such as ultrasensitive response to stimuli. We highlight features of the system relevant for optimizing filter efficiency, encoded in a single, measurable, dimensionless parameter. Our theory, which describes noise control in a large class of signal transduction networks, is also useful both for the design of synthetic biochemical signaling pathways and the manipulation of pathways through experimental probes such as oscillatory input.

We present a thermodynamically robust coarse-grained model to simulate folding of RNA in monovalent salt solutions. The model includes stacking, hydrogen bond, and electrostatic interactions as fundamental components in describing the stability of RNA structures. The stacking interactions are parametrized using a set of nucleotide-specific parameters, which were calibrated against the thermodynamic measurements for single-base stacks and base-pair stacks. All hydrogen bonds are assumed to have the same strength, regardless of their context in the RNA structure. The ionic buffer is modeled implicitly, using the concept of counterion condensation and the Debye-Hückel theory. The three adjustable parameters in the model were determined by fitting the experimental data for two RNA hairpins and a pseudoknot. A single set of parameters provides good agreement with thermodynamic data for the three RNA molecules over a wide range of temperatures and salt concentrations. In the process of calibrating the model, we establish the extent of counterion condensation onto the single-stranded RNA backbone. The reduced backbone charge is independent of the ionic strength and is 60% of the RNA bare charge at 37 °C. Our model can be used to predict the folding thermodynamics for any RNA molecule in the presence of monovalent ions.

We solve a two-dimensional model for polymer chain folding in the presence of mechanical pulling force (f) exactly using equilibrium statistical mechanics. Using analytically derived expression for the partition function we determine the phase diagram for the model in the f-temperature (T) plane. A square root singularity in the susceptibility indicates a second order phase transition from a folded to an unfolded state at a critical force (fc) in the thermodynamic limit of infinitely long polymer chain. The temperature dependence of fc shows a reentrant phase transition, which is reflected in an increase in fc as T increases below a threshold value. As a result, for a range of f values, the unfolded state is stable at both low and high temperatures. The high temperature unfolded state is stabilized by entropy whereas the low temperature unfolded state is dominated by favorable energy. The exact calculation could serve as a benchmark for testing approximate theories that are used in analyzing single molecule pulling experiments.

Using theoretical arguments and extensive Monte Carlo (MC) simulations of a coarse-grained three-dimensional off-lattice model of a β-hairpin, we demonstrate that the equilibrium critical force, Fc, needed to unfold the biopolymer increases nonlinearly with increasing volume fraction occupied by the spherical macromolecular crowding agent. Both scaling arguments and MC simulations show that the critical force increases as Fc ≈ φc(α). The exponent α is linked to the Flory exponent relating the size of the unfolded state of the biopolymer and the number of amino acids. The predicted power law dependence is confirmed in simulations of the dependence of the isothermal extensibility and the fraction of native contacts on φc. We also show using MC simulations that Fc is linearly dependent on the average osmotic pressure (P) exerted by the crowding agents on the β-hairpin. The highly significant linear correlation coefficient of 0.99657 between Fc and P makes it straightforward to predict the dependence of the critical force on the density of crowders. Our predictions are amenable to experimental verification using laser optical tweezers.

In single-molecule laser optical tweezer (LOT) pulling experiments, a protein or RNA is juxtaposed between DNA handles that are attached to beads in optical traps. The LOT generates folding trajectories under force in terms of time-dependent changes in the distance between the beads. How to construct the full intrinsic folding landscape (without the handles and beads) from the measured time series is a major unsolved problem. By using rigorous theoretical methods–which account for fluctuations of the DNA handles, rotation of the optical beads, variations in applied tension due to finite trap stiffness, as well as environmental noise and limited bandwidth of the apparatus–we provide a tractable method to derive intrinsic free-energy profiles. We validate the method by showing that the exactly calculable intrinsic free-energy profile for a generalized Rouse model, which mimics the two-state behavior in nucleic acid hairpins, can be accurately extracted from simulated time series in a LOT setup regardless of the stiffness of the handles. We next apply the approach to trajectories from coarse-grained LOT molecular simulations of a coiled-coil protein based on the GCN4 leucine zipper and obtain a free-energy landscape that is in quantitative agreement with simulations performed without the beads and handles. Finally, we extract the intrinsic free-energy landscape from experimental LOT measurements for the leucine zipper.

It is established that prion protein is the sole causative agent in a number of diseases in humans and animals. However, the nature of conformational changes that the normal cellular form, PrP(C), undergoes in its conversion to a self-replicating state is still not fully understood. The ordered C-terminus of PrP(C) proteins has three helices (H1-H3). Here, we use statistical coupling analysis (SCA) to infer covariations at various locations using a family of evolutionarily related sequences and the response of mouse and human PrP(C)s to mechanical force to decipher the initiation sites for the transition from PrP(C) to an aggregation-prone PrP* state. Sequence-based SCA predicts that the clustered residues in nonmammals are localized in the stable core (near H1) of PrP(C), whereas in mammalian PrP(C), they are localized in frustrated helices H2 and H3 where most of the pathogenic mutations are found. Force-extension curves and free energy profiles as a function of extension of mouse and human PrP(C) in the absence of a disulfide (SS) bond between residues Cys179 and Cys214, generated by applying mechanical force to the ends of the molecule, show a sequence of unfolding events starting first with rupture of H2 and H3. This is followed by disruption of structure in two strands. Helix H1, stabilized by three salt bridges, resists substantial force before unfolding. Force extension profiles and the dynamics of rupture of tertiary contacts also show that even in the presence of an SS bond the instabilities in most of H3 and parts of H2 still determine the propensity to form the PrP* state. In mouse PrP(C) with an SS bond, there are ∼10 residues that retain their order even at high forces. Both SCA and single-molecule force simulations show that in the conversion from PrP(C) to PrP(SC) major conformational changes occur (at least initially) in H2 and H3, which because of their sequence compositions are frustrated in the helical state. Implications of our findings for the structural model for the scrapie form of PrP(C) are discussed.

A quantitative theory of protein folding should make testable predictions using theoretical models and simulations performed under conditions that closely mimic those used in experiments. Typically, in laboratory experiments folding or unfolding is initiated using denaturants or external mechanical force, whereas theories and simulations use temperature as the control parameter, thus making it difficult to make direct comparisons with experiments. The molecular transfer model (MTM), which incorporates environmental changes using measured quantities in molecular simulations, overcomes these difficulties. Predictions of the folding thermodynamics and kinetics of a number of proteins using MTM simulations are in remarkable agreement with experiments. The MTM and all atom simulations demonstrating the presence of dry globules represent major advances in the proteins folding field.

The molecular motor myosin V (MyoV) exhibits a wide repertoire of pathways during the stepping process, which is intimately connected to its biological function. The best understood of these is the hand-over-hand stepping by a swinging lever arm movement toward the plus end of actin filaments. Single-molecule experiments have also shown that the motor “foot stomps,” with one hand detaching and rebinding to the same site, and back-steps under sufficient load. The complete taxonomy of MyoV’s load-dependent stepping pathways, and the extent to which these are constrained by motor structure and mechanochemistry, are not understood. Using a polymer model, we develop an analytical theory to describe the minimal physical properties that govern motor dynamics. We solve the first-passage problem of the head reaching the target-binding site, investigating the competing effects of backward load, strain in the leading head biasing the diffusion in the direction of the target, and the possibility of preferential binding to the forward site due to the recovery stroke. The theory reproduces a variety of experimental data, including the power stroke and slow diffusive search regimes in the mean trajectory of the detached head, and the force dependence of the forward-to-backward step ratio, run length, and velocity. We derive a stall force formula, determined by lever arm compliance and chemical cycle rates. By exploring the MyoV design space, we predict that it is a robust motor whose dynamical behavior is not compromised by reasonable perturbations to the reaction cycle and changes in the architecture of the lever arm.

As a consequence of the rugged landscape of RNA molecules their folding is described by the kinetic partitioning mechanism according to which only a small fraction (φF) reaches the folded state while the remaining fraction of molecules is kinetically trapped in misfolded intermediates. The transition from the misfolded states to the native state can far exceed biologically relevant time. Thus, RNA folding in vivo is often aided by protein cofactors, called RNA chaperones, that can rescue RNAs from a multitude of misfolded structures. We consider two models, based on chemical kinetics and chemical master equation, for describing assisted folding. In the passive model, applicable for class I substrates, transient interactions of misfolded structures with RNA chaperones alone are sufficient to destabilize the misfolded structures, thus entropically lowering the barrier to folding. For this mechanism to be efficient the intermediate ribonucleoprotein complex between collapsed RNA and protein cofactor should have optimal stability. We also introduce an active model (suitable for stringent substrates with small φF), which accounts for the recent experimental findings on the action of CYT-19 on the group I intron ribozyme, showing that RNA chaperones do not discriminate between the misfolded and the native states. In the active model, the RNA chaperone system utilizes chemical energy of adenosine triphosphate hydrolysis to repeatedly bind and release misfolded and folded RNAs, resulting in substantial increase of yield of the native state. The theory outlined here shows, in accord with experiments, that in the steady state the native state does not form with unit probability.

Riboswitches are RNA elements that allosterically regulate gene expression by binding cellular metabolites. The SAM-III riboswitch, one of several classes that binds S-adenosylmethionine (SAM), represses translation upon binding SAM (OFF state) by encrypting the ribosome binding sequence. We have carried out simulations of the RNA by applying mechanical force (f) to the ends of SAM-III, with and without SAM, to get quantitative insights into the f-dependent structural changes. Force-extension (z) curves (FECs) for the apo (ON) state, obtained in simulations in which f is increased at a constant loading rate, show three intermediates, with the first one being the rupture of SAM binding region, which is greatly stabilized in the OFF state. Force-dependent free energy profiles, G(z,f), as a function of z, obtained in equilibrium constant force simulations, reveal the intermediates observed in FECs. The predicted stability difference between the ON and OFF states using G(z,f) is in excellent agreement with experiments. Remarkably, using G(z,f)s and estimate of an effective diffusion constant at a single value of f allows us to predict the f-dependent transition rates using theory of first passage times for both the apo and holo states. To resolve the kinetics of assembly of SAM-III riboswitch in structural terms, we use force stretch-quench pulse sequences in which the force on RNA is maintained at a low (fq) value starting from a high value for a time period tq. Variation of tq over a wide range results in resolution of elusive states involved in the SAM binding pocket and leads to accurate determination of folding times down to fq = 0. Quantitative measure of the folding kinetics, obtained from the folding landscape, allows us to propose that, in contrast to riboswitches regulating transcription, SAM-III functions under thermodynamic control provided the basal concentration of SAM exceeds a small critical value. All of the predictions are amenable to tests in single molecule pulling experiments.

We use Brownian dynamics simulations of a binary mixture of highly charged spherical colloidal particles to test some of the predictions of the random first-order transition (RFOT) theory [Phys. Rev. Lett. 58, 2091 (1987); Phys. Rev. A 40, 1045 (1989)]. In accord with mode-coupling theory and RFOT, we find that as the volume fraction of the colloidal particles ϕ approaches the dynamical transition value ϕ(A), three measures of dynamics show an effective ergodic to nonergodic transition. First, there is a dramatic slowing down of diffusion, with the translational diffusion constant decaying as a power law as ϕ→ϕ(A)(-). Second, the energy metric, a measure of ergodicity breaking in classical many-body systems, shows that the system becomes effectively nonergodic as ϕ(A) is approached. Finally, the time t(*), at which the four-point dynamical susceptibility achieves a maximum, also increases as a power law near ϕ(A). Remarkably, the translational diffusion coefficients, ergodic diffusion coefficient, and (t(*))(-) all vanish as (ϕ(-1)-ϕ(A)(-1))(γ) with both ϕ(A)(≈0.1) and γ being the roughly the same for all three quantities. Above ϕ(A), transport involves crossing free energy barriers. In this regime, the density-density correlation function decays as a stretched exponential [exp-(t/τ(α))(β)] with β≈0.45. The ϕ dependence of the relaxation time τ(α) could be fit using the Vogel-Tamman-Fulcher law with the ideal glass transition at ϕ(K)≈0.47. By using a local entropy measure, we show that the law of large numbers is not obeyed above ϕ(A), and gives rise to subsample to subsample fluctuations in all physical observables. We propose that dynamical heterogeneity is a consequence of violation of law of large numbers.

Chemical physics as a discipline contributes many experimental tools, algorithms, and fundamental theoretical models that can be applied to biological problems. This is especially true now as the molecular level and the systems level descriptions begin to connect, and multi-scale approaches are being developed to solve cutting edge problems in biology. In some cases, the concepts and tools got their start in non-biological fields, and migrated over, such as the idea of glassy landscapes, fluorescence spectroscopy, or master equation approaches. In other cases, the tools were specifically developed with biological physics applications in mind, such as modeling of single molecule trajectories or super-resolution laser techniques. In this introduction to the special topic section on chemical physics of biological systems, we consider a wide range of contributions, all the way from the molecular level, to molecular assemblies, chemical physics of the cell, and finally systems-level approaches, based on the contributions to this special issue. Chemical physicists can look forward to an exciting future where computational tools, analytical models, and new instrumentation will push the boundaries of biological inquiry.

The osmolyte trimethylamine N-oxide (TMAO) stabilizes the tertiary but not the secondary structures of RNA. However, molecular dynamics simulations performed on the PreQ1 riboswitch showed that TMAO destabilizes the tertiary riboswitch structure, leading us to hypothesize that the presence of RNA could result in enhanced population of the protonated form, TMAOP. Constant pH replica exchange simulations showed that a percentage of TMAO is indeed protonated, thus contributing to the stability of the tertiary but not the secondary structure of PreQ1. TMAOP results in an unfavorable dehydration of phosphodiester backbone, which is compensated by electrostatic attraction between TMAOP and the phosphate groups. In addition, TMAOP interacts with specific sites in the tertiary RNA structure, mimicking the behavior of positively charged ions and of the PreQ1 ligand in stabilizing RNA. Finally, we predict that TMAO-induced stabilization of RNA tertiary structures should be strongly pH dependent

Urea, a polar molecule with a large dipole moment, not only destabilizes folded RNA structures but can also enhance the folding rates of large ribozymes. Unlike the mechanism of urea-induced unfolding of proteins, which is well understood, the action of urea on RNA has barely been explored. We performed extensive all-atom molecular dynamics simulations to determine the molecular underpinnings of urea-induced RNA denaturation. Urea displays its denaturing power in both secondary and tertiary motifs of the riboswitch structure. Our simulations reveal that the denaturation of RNA structures is mainly driven by the hydrogen-bonding and stacking interactions of urea with the bases. Through detailed studies of the simulation trajectories, we found that geminate pairs between urea and bases due to hydrogen bonds and stacks persist only ~0.1-1 ns, which suggests that the urea-base interaction is highly dynamic. Most importantly, the early stage of base-pair disruption is triggered by penetration of water molecules into the hydrophobic domain between the RNA bases. The infiltration of water into the narrow space between base pairs is critical in increasing the accessibility of urea to transiently disrupted bases, thus allowing urea to displace inter-base hydrogen bonds. This mechanism–water-induced disruption of base pairs resulting in the formation of a “wet” destabilized RNA followed by solvation by urea–is the exact opposite of the two-stage denaturation of proteins by urea. In the latter case, initial urea penetration creates a dry globule, which is subsequently solvated by water, leading to global protein unfolding. Our work shows that the ability to interact with both water and polar or nonpolar components of nucleotides makes urea a powerful chemical denaturant for nucleic acids.

A variety of neurodegenerative diseases are associated with amyloid plaques, which begin as soluble protein oligomers but develop into amyloid fibrils. Our incomplete understanding of this process underscores the need to decipher the principles governing protein aggregation. Mechanisms of in vivo amyloid formation involve a number of coconspirators and complex interactions with membranes. Nevertheless, understanding the biophysical basis of simpler in vitro amyloid formation is considered important for discovering ligands that preferentially bind regions harboring amyloidogenic tendencies. The determination of the fibril structure of many peptides has set the stage for probing the dynamics of oligomer formation and amyloid growth through computer simulations. Most experimental and simulation studies, however, have been interpreted largely from the perspective of proteins: the role of solvent has been relatively overlooked in oligomer formation and assembly to protofilaments and amyloid fibrils. In this Account, we provide a perspective on how interactions with water affect folding landscapes of amyloid beta (Aβ) monomers, oligomer formation in the Aβ16-22 fragment, and protofilament formation in a peptide from yeast prion Sup35. Explicit molecular dynamics simulations illustrate how water controls the self-assembly of higher order structures, providing a structural basis for understanding the kinetics of oligomer and fibril growth. Simulations show that monomers of Aβ peptides sample a number of compact conformations. The formation of aggregation-prone structures (N*) with a salt bridge, strikingly similar to the structure in the fibril, requires overcoming a high desolvation barrier. In general, sequences for which N* structures are not significantly populated are unlikely to aggregate. Oligomers and fibrils generally form in two steps. First, water is expelled from the region between peptides rich in hydrophobic residues (for example, Aβ16-22), resulting in disordered oligomers. Then the peptides align along a preferred axis to form ordered structures with anti-parallel β-strand arrangement. The rate-limiting step in the ordered assembly is the rearrangement of the peptides within a confining volume. The mechanism of protofilament formation in a polar peptide fragment from the yeast prion, in which the two sheets are packed against each other and create a dry interface, illustrates that water dramatically slows self-assembly. As the sheets approach each other, two perfectly ordered one-dimensional water wires form. They are stabilized by hydrogen bonds to the amide groups of the polar side chains, resulting in the formation of long-lived metastable structures. Release of trapped water from the pore creates a helically twisted protofilament with a dry interface. Similarly, the driving force for addition of a solvated monomer to a preformed fibril is water release; the entropy gain and favorable interpeptide hydrogen bond formation compensate for entropy loss in the peptides. We conclude by offering evidence that a two-step model, similar to that postulated for protein crystallization, must also hold for higher order amyloid structure formation starting from N*. Distinct water-laden polymorphic structures result from multiple N* structures. Water plays multifarious roles in all of these protein aggregations. In predominantly hydrophobic sequences, water accelerates fibril formation. In contrast, water-stabilized metastable intermediates dramatically slow fibril growth rates in hydrophilic sequences.

We show that the folding rates (k(F)s) of RNA are determined by N, the number of nucleotides. By assuming that the distribution of free-energy barriers separating the folded and the unfolded states is Gaussian, which follows from central limit theorem arguments and polymer physics concepts, we show that k(F)≈k(0)exp(-αN(0.5)). Remarkably, the theory fits experimental rates spanning over 7 orders of magnitude with k(0)~1.0(μs)(-1). Our finding suggests that the speed limit of RNA folding is ~ 1 μs, [corrected] just as it is in the folding of globular proteins.

We use molecular simulations using a coarse-grained model to map the folding landscape of Green Fluorescent Protein (GFP), which is extensively used as a marker in cell biology and biotechnology. Thermal and Guanidinium chloride (GdmCl) induced unfolding of a variant of GFP, without the chromophore, occurs in an apparent two-state manner. The calculated midpoint of the equilibrium folding in GdmCl, taken into account using the Molecular Transfer Model (MTM), is in excellent agreement with the experiments. The melting temperatures decrease linearly as the concentrations of GdmCl and urea are increased. The structural features of rarely populated equilibrium intermediates, visible only in free energy profiles projected along a few order parameters, are remarkably similar to those identified in a number of ensemble experiments in GFP with the chromophore. The excellent agreement between simulations and experiments show that the equilibrium intermediates are stabilized by the chromophore. Folding kinetics, upon temperature quench, show that GFP first collapses and populates an ensemble of compact structures. Despite the seeming simplicity of the equilibrium folding, flux to the native state flows through multiple channels and can be described by the kinetic partitioning mechanism. Detailed analysis of the folding trajectories show that both equilibrium and several kinetic intermediates, including misfolded structures, are sampled during folding. Interestingly, the intermediates characterized in the simulations coincide with those identified in single molecule pulling experiments. Our predictions, amenable to experimental tests, show that MTM is a practical way to simulate the effect of denaturants on the folding of large proteins.

Kinesin walks processively on microtubules in an asymmetric hand-over-hand manner with each step spanning 16 nm. We used molecular simulations to determine the fraction of a single step due to conformational changes in the neck linker, and that due to diffusion of the tethered head. Stepping is determined largely by two energy scales, one favoring neck-linker docking and the other, ε(h)(MT-TH), between the trailing head (TH) and the microtubule. Neck-linker docking and an optimal value of ε(h)(MT-TH) are needed to minimize the probability that the TH takes side steps. There are three major stages in the kinematics of a step. In the first, the neck linker docks, resulting in ∼(5-6) nm movements of the trailing head. The TH moves an additional (6-8) nm in stage II by anisotropic translational diffusion. In the third stage, spanning ∼(3-4) nm, the step is complete with the TH binding to the αβ-tubulin binding site.

Protein conformations change among distinct thermodynamic states as solution conditions (temperature, denaturants, pH) are altered or when they are subjected to mechanical forces. A quantitative description of the changes in the relative stabilities of the various thermodynamic states is needed to interpret and predict experimental outcomes. We provide a framework based on the Molecular Transfer Model (MTM) to account for pH effects on the properties of globular proteins. The MTM utilizes the partition function of a protein calculated from molecular simulations at one set of solution conditions to predict protein properties at another set of solution conditions. To take pH effects into account, we utilized experimentally measured pK(a) values in the native and unfolded states to calculate the free energy of transferring a protein from a reference pH to the pH of interest. We validate our approach by demonstrating that the native-state stability as a function of pH is accurately predicted for chymotrypsin inhibitor 2 (CI2) and protein G. We use the MTM to predict the response of CI2 and protein G subjected to a constant force (f) and varying pH. The phase diagrams of CI2 and protein G as a function of f and pH are dramatically different and reflect the underlying pH-dependent stability changes in the absence of force. The calculated equilibrium free energy profiles as functions of the end-to-end distance of the two proteins show that, at various pH values, CI2 unfolds via an intermediate when subjected to f. The locations of the two transition states move toward the more unstable state as f is changed, which is in accord with the Hammond-Leffler postulate. In sharp contrast, force-induced unfolding of protein G occurs in a single step. Remarkably, the location of the transition state with respect to the folded state is independent of f, which suggests that protein G is mechanically brittle. The MTM provides a natural framework for predicting the outcomes of ensemble and single-molecule experiments for a wide range of solution conditions.

Folding mechanisms in which secondary structures are stabilized through the formation of tertiary interactions are well documented in protein folding but challenge the folding hierarchy normally assumed for RNA. However, it is increasingly clear that RNA could fold by a similar mechanism. P5abc, a small independently folding tertiary domain of the Tetrahymena thermophila group I ribozyme, is known to fold by a secondary structure rearrangement involving helix P5c. However, the extent of this rearrangement and the precise stage of folding that triggers it are unknown. We use experiments and simulations to show that the P5c helix switches to the native secondary structure late in the folding pathway and is directly coupled to the formation of tertiary interactions in the A-rich bulge. P5c mutations show that the switch in P5c is not rate-determining and suggest that non-native interactions in P5c aid folding rather than impede it. Our study illustrates that despite significant differences in the building blocks of proteins and RNA, there may be common ways in which they self-assemble.

We investigate the structural transitions in a polymer induced by spherical and nonspherical crowding particles over a wide range of conditions. The polymer conformations are specified by the radius of gyration and the quality of the solvent in the absence of crowding particles. In the presence of crowding particles, the structures are altered by the volume fraction, size, shape, and polydispersity of the crowders. We show that crowding induces an array of structural changes, ranging from helix, helical hairpin (HH), and multiple helix bundles (HBs), depending on the interplay of multiple length and energy scales including the solvent quality, length of the polymer, temperature, and the characteristics of the crowding agents. In nearly good solvents, the polymer undergoes coil-helix transition in accord with the predictions based on the entropic stabilization mechanism. Higher-order (HH and HB) structures are obtained in poor or moderately poor solvents. In a binary mixture of spherical crowders, the effect of the two components is largely additive with the polymer undergoing greater compaction at higher volume fraction. In contrast to spherical crowders, spherocylinder-like crowders have a dramatically different effect on the diagram of states of the polymer. In the presence of spherocylinders, the polymer prefers to form a nearly ideal helix, especially at low temperatures and high aspect ratios of the crowders, at volume fractions that are not large enough for nematic order. Surprisingly, there is a complete absence of HH and HB in the range of conditions explored here. The dominant formation of spherocylinder-induced helix formation is due to the tendency of the spherocylinders and the polymer to align along the director formed by an increase in nematic order only in the vicinity of the polymer. Our study, which has produced several testable predictions, shows that only by probing the effects of crowding on a polymer (or a protein and RNA) over a wide range of conditions can the diagram of states be quantitatively described.

Force-extension curves (FECs), which quantify the response of a variety of biomolecules subject to mechanical force (f), are often quantitatively fit using worm-like chain (WLC) or freely jointed chain (FJC) models. These models predict that the chain extension, x, normalized by the contour length increases linearly at small f and at high forces scale as x ~ (1 – f(-α)), where α = 0.5 for WLC and unity for FJC. In contrast, experiments on single-stranded DNA (ssDNA) show that over a range of f and ionic concentration, x scales as x ~ ln f, which cannot be explained using WLC or FJC models. Using theory and simulations we show that this unusual behavior in FEC in ssDNA is due to sequence-independent polyelectrolyte effects. We show that the x ~ ln f arises because in the absence of force the tangent correlation function, quantifying chain persistence, decays algebraically on length scales on the order of the Debye length. Our theory, which is most appropriate for monovalent salts, quantitatively fits the experimental data and further predicts that such a regime is not discernible in double-stranded DNA.

A theoretical basis for the molecular transfer model (MTM), which takes into account the effects of denaturants by combining experimental data and molecular models for proteins, is provided. We show that the MTM is a mean field-like model that implicitly takes into account denaturant-induced many body interactions. The MTM in conjunction with the coarse-grained self organized polymer model with side chains (SOP-SC) for polypeptide chains is used to simulate the folding of the src-SH3 domain as a function of temperature (T) and guanidine hydrochloride (GdmCl) concentration [C]. Besides reproducing the thermodynamic aspects of SH3 folding, the SOP-SC also captures the cooperativity of the folding transitions. A number of experimentally testable predictions are also made. First, we predict that the melting temperature T(m)([C]) decreases linearly as [C] increases. Second, we show that the midpoints C(m,i) and melting temperatures T(m,i) at which individual residues acquire 50% of their native contacts differ from the global midpoint (C(m) ≈ 2.5 M) and melting temperature (T(m) = 355 K) at which the folded and unfolded states coexist. Dispersion in C(m,i) is greater than that found for T(m,i). Third, folding kinetics at [C] = 0 M shows that the acquisition of contacts between all the secondary structural elements and global folding occur nearly simultaneously. Finally, from the free energy profiles as a function of the structural overlap function and the radius of gyration of the protein, we find that at a fixed T the transition state moves toward the folded state as [C] increases in accord with the Hammond postulate. In contrast, we predict that along the locus of points T(m)([C]) the location of the transition state does not change. The theory and the models used here are sufficiently general for studying the folding of other single domain proteins.

Riboswitches, structured elements in the untranslated regions of messenger RNAs, regulate gene expression by binding specific metabolites. We introduce a kinetic network model that describes the functions of riboswitches at the systems level. Using experimental data for flavin mononucleotide riboswitch as a guide, we show that efficient function, implying a large dynamic range without compromising the requirement to suppress transcription, is determined by a balance between the transcription speed, the folding and unfolding rates of the aptamer, and the binding rates of the metabolite. We also investigated the effect of negative feedback accounting for binding to metabolites, which are themselves the products of genes that are being regulated. For a range of transcription rates negative feedback suppresses gene expression by nearly 10-fold. Negative feedback speeds the gene expression response time, and suppresses the change of steady-state protein concentration by half relative to that without feedback, when there is a modest spike in DNA concentration. A dynamic phase diagram expressed in terms of transcription speed, folding rates, and metabolite binding rates predicts different scenarios in riboswitch-mediated transcription regulation.

A plausible consequence of the rugged folding energy landscapes inherent to biomolecules is that there may be more than one functionally competent folded state. Indeed, molecule-to-molecule variations in the folding dynamics of enzymes and ribozymes have recently been identified in single-molecule experiments, but without systematic quantification or an understanding of their structural origin. Here, using concepts from glass physics and complementary clustering analysis, we provide a quantitative method to analyse single-molecule fluorescence resonance energy transfer (smFRET) data, thereby probing the isomerization dynamics of Holliday junctions, which display such heterogeneous dynamics over a long observation time (T(obs) ≈ 40 s). We show that the ergodicity of Holliday junction dynamics is effectively broken and that their conformational space is partitioned into a folding network of kinetically disconnected clusters. Theory suggests that the persistent heterogeneity of Holliday junction dynamics is a consequence of internal multiloops with varying sizes and flexibilities frozen by Mg(2+) ions. An annealing experiment using Mg(2+) pulses lends support to this idea by explicitly showing that interconversions between trajectories with different patterns can be induced.

Curvatures in the most probable rupture force (f*) versus log-loading rate (log r(f)) observed in dynamic force spectroscopy (DFS) on biomolecular complexes are interpreted using a one-dimensional free energy profile with multiple barriers or a single barrier with force-dependent transition state. Here, we provide a criterion to select one scenario over another. If the rupture dynamics occurs by crossing a single barrier in a physical free energy profile describing unbinding, the exponent ν, from (1 – f*/f(c))(1/ν) ~ (log r(f)) with f(c) being a critical force in the absence of force, is restricted to 0.5 ≤ ν ≤ 1. For biotin-ligand complexes and leukocyte-associated antigen-1 bound to intercellular adhesion molecules, which display large curvature in the DFS data, fits to experimental data yield ν < 0.5, suggesting that if ligand unbinding is assumed to proceed along one-dimensional pulling coordinate, the dynamics should occur in a energy landscape with multiple-barriers.

Non-coding RNA sequences play a great role in controlling a number of cellular functions, thus raising the need to understand their complex conformational dynamics in quantitative detail. In this perspective, we first show that single molecule pulling when combined with with theory and simulations can be used to quantitatively explore the folding landscape of nucleic acid hairpins, and riboswitches with tertiary interactions. Applications to riboswitches, which are non-coding RNA elements that control gene expression by undergoing dynamical conformational changes in response to binding of metabolites, lead to an organization principle that assembly of RNA is determined by the stability of isolated helices. We also point out the limitations of single molecule pulling experiments, with molecular extension as the only accessible parameter, in extracting key parameters of the folding landscapes of RNA molecules.

The distances over which biological molecules and their complexes can function range from a few nanometres, in the case of folded structures, to millimetres, for example, during chromosome organization. Describing phenomena that cover such diverse length, and also time, scales requires models that capture the underlying physics for the particular length scale of interest. Theoretical ideas, in particular, concepts from polymer physics, have guided the development of coarse-grained models to study folding of DNA, RNA and proteins. More recently, such models and their variants have been applied to the functions of biological nanomachines. Simulations using coarse-grained models are now poised to address a wide range of problems in biology.

Quantitative description of how proteins fold under experimental conditions remains a challenging problem. Experiments often use urea and guanidinium chloride to study folding whereas the natural variable in simulations is temperature. To bridge the gap, we use the molecular transfer model that combines measured denaturant-dependent transfer free energies for the peptide group and amino acid residues, and a coarse-grained C(α)-side chain model for polypeptide chains to simulate the folding of src SH(3) domain. Stability of the native state decreases linearly as [C] (the concentration of guanidinium chloride) increases with the slope, m, that is in excellent agreement with experiments. Remarkably, the calculated folding rate at [C] = 0 is only 16-fold larger than the measured value. Most importantly ln k(obs) (k(obs) is the sum of folding and unfolding rates) as a function of [C] has the characteristic V (chevron) shape. In every folding trajectory, the times for reaching the native state, interactions stabilizing all the substructures, and global collapse coincide. The value of (m(f) is the slope of the folding arm of the chevron plot) is identical to the fraction of buried solvent accessible surface area in the structures of the transition state ensemble. In the dominant transition state, which does not vary significantly at low [C], the core of the protein and certain loops are structured. Besides solving the long-standing problem of computing the chevron plot, our work lays the foundation for incorporating denaturant effects in a physically transparent manner either in all-atom or coarse-grained simulations.

We establish a framework for assessing whether the transition state location of a biopolymer, which can be inferred from single molecule pulling experiments, corresponds to the ensemble of structures that have equal probability of reaching either the folded or unfolded states (P(fold)=0.5). Using results for the forced unfolding of a RNA hairpin, an exactly soluble model, and an analytic theory, we show that P(fold) is solely determined by s, an experimentally measurable molecular tensegrity parameter, which is a ratio of the tensile force and a compaction force that stabilizes the folded state. Applications to folding landscapes of DNA hairpins and a leucine zipper with two barriers provide a structural interpretation of single molecule experimental data. Our theory can be used to assess whether molecular extension is a good reaction coordinate using measured free energy profiles.

Formation of a pseudoknot (PK) in the conserved RNA core domain in the ribonucleoprotein human telomerase is required for function. In vitro experiments show that the PK is in equilibrium with an extended hairpin (HP) structure. We use molecular simulations of a coarse-grained model, which reproduces most of the salient features of the experimental melting profiles of PK and HP, to show that crowding enhances the stability of PK relative to HP in the wild type and in a mutant associated with dyskeratosis congenita. In monodisperse suspensions, small crowding particles increase the stability of compact structures to a greater extent than larger crowders. If the sizes of crowders in a binary mixture are smaller than that of the unfolded RNA, the increase in melting temperature due to the two components is additive. In a ternary mixture of crowders that are larger than the unfolded RNA, which mimics the composition of ribosome, large enzyme complexes and proteins in Escherichia coli , the marginal increase in stability is entirely determined by the smallest component. We predict that crowding can partially restore telomerase activity in mutants with decreased PK stability.

The osmolyte trimethylamine N-oxide (TMAO) accumulates in the cell in response to osmotic stress and increases the thermodynamic stability of folded proteins. To understand the mechanism of TMAO induced stabilization of folded protein states, we systematically investigated the action of TMAO on several model dipeptides (leucine, L(2), serine, S(2), glutamine, Q(2), lysine, K(2), and glycine, G(2)) in order to elucidate the effect of residue-specific TMAO interactions on small fragments of solvent-exposed conformations of the denatured states of proteins. We find that TMAO preferentially hydrogen bonds with the exposed dipeptide backbone but generally not with nonpolar or polar side chains. However, interactions with the positively charged Lys are substantially greater than with the backbone. The dipeptide G(2) is a useful model of the pure amide backbone; interacts with TMAO by forming a hydrogen bond between the amide nitrogen and the oxygen in TMAO. In contrast, TMAO is depleted from the protein backbone in the hexapeptide G(6), which shows that the length of the polypeptide chain is relevant in aqueous TMAO solutions. These simulations lead to the hypothesis that TMAO-induced stabilization of proteins and peptides is a consequence of depletion of the solute from the protein surface provided intramolecular interactions are more favorable than those between TMAO and the backbone. To test our hypothesis, we performed additional simulations of the action of TMAO on an intrinsically disordered Aβ(16-22) (KLVFFAE) monomer. In the absence of TMAO, Aβ(16-22) is a disordered random coil. However, in aqueous TMAO solution, Aβ(16-22) monomer samples compact conformations. A transition from random coil to α-helical secondary structure is observed at high TMAO concentrations. The coil to α-helix transition is highly cooperative especially considering the small number of residues in Aβ(16-22). Our work highlights the potential similarities between the action of TMAO on long polypeptide chains and entropic stabilization of proteins in a crowded environment due to excluded volume interactions. In this sense, the chemical chaperone TMAO is a nanocrowding particle.

Understanding the influence of macromolecular crowding and nanoparticles on the formation of in-register β-sheets, the primary structural component of amyloid fibrils, is a first step towards describing in vivo protein aggregation and interactions between synthetic materials and proteins. Using all atom molecular simulations in implicit solvent we illustrate the effects of nanoparticle size, shape, and volume fraction on oligomer formation of an amyloidogenic peptide from the transthyretin protein. Surprisingly, we find that inert spherical crowding particles destabilize in-register β-sheets formed by dimers while stabilizing β-sheets comprised of trimers and tetramers. As the radius of the nanoparticle increases crowding effects decrease, implying smaller crowding particles have the largest influence on the earliest amyloid species. We explain these results using a theory based on the depletion effect. Finally, we show that spherocylindrical crowders destabilize the ordered β-sheet dimer to a greater extent than spherical crowders, which underscores the influence of nanoparticle shape on protein aggregation.

Single molecule experiments that initiate folding using mechanical force are uniquely suited to reveal the nature of populated states in the folding process. Using a strategy proposed on theoretical grounds, which calls for repeated cycling of force from high to low values using force pulses, it was demonstrated in atomic force spectroscopy (AFM) experiments that an ensemble of minimum energy compact structures (MECS) are sampled during the folding of polyubiquitin. The structures in the ensemble are mechanically resistant to a lesser extent than the native state. Remarkably, forced unfolding of the populated intermediates reveals a broad distribution of extensions including steps up to 30 nm and beyond. We show using molecular simulations that favorable interdomain interactions leading to domain swapping between adjacent ubiquitin modules results in the formation of the ensemble of MECS, whose unfolding leads to an unusually broad distribution of steps. We obtained the domain-swapped structures using coarse-grained ubiquitin dimer models by exchanging native interactions between two monomeric ubiquitin molecules. Brownian dynamics force unfolding of the proposed domain-swapped structures, with mechanical stability that is approximately 100-fold lower than the native state, gives rise to a distribution of extensions from 2 to 30 nm. Our results, which are in quantitative agreement with AFM experiments, suggest that domain swapping may be a general mechanism in the assembly of multi-sub-unit proteins.

Quantitative understanding of the kinetics of fibril formation and the molecular mechanism of transition from monomers to fibrils is needed to obtain insights into the growth of amyloid fibrils and more generally self-assembly multisubunit protein complexes. Significant advances using computations of protein aggregation in a number of systems have established generic and sequence-specific aspects of the early steps in oligomer formation. Theoretical considerations, which view oligomer and fibril growth as diffusion in a complex energy landscape, and computational studies, involving minimal lattice and coarse-grained models, have revealed general principles governing the transition from monomeric protein to ordered fibrillar aggregates. Detailed atomistic calculations have explored the early stages of the protein aggregation pathway for a number of amyloidogenic proteins, most notably amyloid β- (Aβ-) protein and fragments from proteins linked to various diseases. These computational studies have provided insights into the role of sequence, role of water, and specific interatomic interactions underlying the thermodynamics and dynamics of elementary kinetic steps in the aggregation pathway. Novel methods are beginning to illustrate the structural basis for the production of Aβ-peptides through interactions with secretases in the presence of membranes. We show that a variety of theoretical approaches, ranging from scaling arguments to minimal models to atomistic simulations, are needed as a complement to experimental studies probing the principles governing protein aggregation.

Globally RNA folding occurs in multiple stages involving chain compaction and subsequent rearrangement by a number of parallel routes to the folded state. However, the sequence-dependent details of the folding pathways and the link between collapse and folding are poorly understood. To obtain a comprehensive picture of the thermodynamics and folding kinetics we used molecular simulations of coarse-grained model of a pseudoknot found in the conserved core domain of the human telomerase (hTR) by varying both temperature (T) and ion concentration (C). The phase diagram in the [T,C] plane shows that the boundary separating the folded and unfolded state for the finite 47-nucleotide system is relatively sharp, implying that from a thermodynamic perspective hTR behaves as an apparent two-state system. However, the folding kinetics following single C-jump or T-quench is complicated, involving multiple channels to the native state. Although globally folding kinetics triggered by T-quench and C-jump are similar, the kinetics of chain compaction are vastly different, which reflects the role of initial conditions in directing folding and collapse. Remarkably, even after substantial reduction in the overall size of hTR, the ensemble of compact conformations are far from being nativelike, suggesting that the search for the folded state occurs among the ensemble of low-energy fluidlike globules. The rate of unfolding, which occurs in a single step, is faster upon C-decrease compared to a jump in temperature. To identify “hidden” states that are visited during the folding process we performed simulations by periodically interrupting the approach to the folded state by lowering C. These simulations show that hTR reaches the folded state through a small number of connected clusters that are repeatedly visited during the pulse sequence in which the folding or unfolding is interrupted. The results from interrupted folding simulations, which are in accord with non-equilibrium single-molecule folding of a large ribozyme, show that multiple probes are needed to reveal the invisible states that are sampled by RNA as it folds. Although we have illustrated the complexity of RNA folding using hTR as a case study, general arguments and qualitative comparisons to time-resolved scattering experiments on Azoarcus group I ribozyme and single-molecule non-equilibrium periodic ion-jump experiments establish the generality of our findings.

PURPOSE: The purpose of this study is to validate a folate-receptor (FR)-targeted dendrimer, PEG-G3-(Gd-DTPA)11-(folate)5, for its ability to detect FR-positive tumors, by using dynamic contrast-enhanced MRI. PROCEDURES: KB cells, FR siRNA knockdown KB cells, and FR negative HT-1080 cells, were incubated with fluorescein-labeled dendrimer and their cellular uptake was observed. Dynamic contrast-enhanced MRI was performed on mice-bearing KB and HT-1080 tumors and the enhancement patterns and parameters were analyzed. RESULTS: Green fluorescence was found in the KB cells in the cellular uptake experiment, but was not seen in other settings. In the dynamic contrast-enhanced MRI, the 30-min washout percentage was -4 +/- 18% in the KB tumors and 39 +/- 23% in the HT-1080 tumors. A 17% cut-off point gave a sensitivity of 94.4% and a specificity of 93.8%. CONCLUSIONS: We have demonstrated the targeting ability of PEG-G3-(Gd-DTPA)11-(folate)5 in vitro and in vivo. A 17% cut-off point for a 30-min washout percentage can be a useful parameter for the diagnosis of FR-positive tumors.

Amyloid-like fibrils from a number of small peptides that are unrelated by sequence adopt a cross-β-spine in which the two sheets fully interdigitate to create a dry interface. Formation of such a dry interface is usually associated with self-assembly of extended hydrophobic surfaces. Here we investigate how a dry interface is created in the process of protofilament formation in vastly different sequences using two amyloidogenic peptides, one a polar sequence from the N terminus of the yeast prion Sup35 and the other a predominantly hydrophobic sequence from the C terminus of Aβ-peptide. Using molecular dynamics simulations with three force fields we show that spontaneous formation of two ordered one-dimensional water wires in the pore between the two sheets of the Sup35 protofilaments results in long-lived structures, which are stabilized by a network of hydrogen bonds between the water molecules in the wires and the polar side chains in the β-sheet. Upon decreasing the stability of the metastable structures, water molecules are expelled resulting in a helically twisted protofilament in which side chains from a pair of β-strands in each sheet pack perfectly resulting in a dry interface. Although drying in hydrophobically dominated interfaces is abrupt, resembling a liquid to vapor transition, we find that discrete transitions between the liquid to one-dimensional ordered water in the nanopore enclosed by the two β-sheets to dry interface formation characterizes protofilament assembly in the yeast prions. Indeed, as the two sheets of the hydrophobic Aβ-sequence approach each other, fibril formation and expulsion of water molecules occur rapidly and nearly simultaneously.

Molecular dynamics simulations and single molecule experiments are used to suggest that charged helices in the medial tail domain participate in myosin VI dimerization (Kim et al., 2010), which reinforces the mechanism that unfolding of the three helix bundle in the proximal tail serves as a lever arm extension.

Identifying the principles that describe the formation of protein oligomers and fibrils with distinct morphologies is a daunting problem. Here we summarize general principles of oligomer formation gleaned from molecular dynamics simulations of Abeta-peptides. The spectra of high free energy structures sampled by the monomer provide insights into the plausible fibril structures, providing a rationale for the ‘strain phenomenon.’ Heterogeneous growth dynamics of small oligomers of Abeta(16-22), whose lowest free energy structures are like nematic droplets, can be broadly described using a two-stage dock-lock mechanism. In the growth process, water is found to play various roles depending on the oligomer size, and peptide length, and sequence. Water may be an explicit element of fibril structure linked to various fibril morphologies.

RNA synthesis, carried out by DNA-dependent RNA polymerase (RNAP) in a process called transcription, involves several stages. In bacteria, transcription initiation starts with promoter recognition and binding of RNAP holoenzyme, resulting in the formation of the closed (R.P(c)) RNAP-promoter DNA complex. Subsequently, a transition to the open R.P(o) complex occurs, characterized by separation of the promoter DNA strands in an approximately 12 base-pair region to form the transcription bubble. Using coarse-grained self-organized polymer models of Thermus aquatics RNAP holoenzyme and promoter DNA complexes, we performed Brownian dynamics simulations of the R.P(c) –> R.P(o) transition. In the fast trajectories, unwinding of the promoter DNA begins by local melting around the -10 element, which is followed by sequential unzipping of DNA till the +2 site. The R.P(c) –> R.P(o) transition occurs in three steps. In step I, dsDNA melts and the nontemplate strand makes stable interactions with RNAP. In step II, DNA scrunches into RNA polymerase and the downstream base pairs sequentially open to form the transcription bubble, which results in strain build up. Subsequently, downstream dsDNA bending relieves the strain as R.P(o) forms. Entry of the dsDNA into the active-site channel of RNAP requires widening of the channel, which occurs by a swing mechanism involving transient movements of a subdomain of the beta subunit caused by steric repulsion with the DNA template strand. If premature local melting away from the -10 element occurs first then the transcription bubble formation is slow involving reformation of the opened base pairs and subsequent sequential unzipping as in the fast trajectories.

The detachment kinetics from actin upon ATP binding is a key step in the reaction cycle of myosin V. We show that a network of residues, constituting the allostery wiring diagram (AWD), that trigger the rigor (R) to post-rigor (PR) transition, span key structural elements from the ATP and actin-binding regions. Several of the residues are in the 33 residue helix (H18), P loop, and switch I. Brownian dynamics simulations show that a hierarchy of kinetically controlled local structural changes leads to the opening of the “cleft” region, resulting in the detachment of the motor domain from actin. Movements in switch I and P loop facilitate changes in the rest of the motor domain, in particular the rotation of H18, whose stiffness within the motor domain is crucial in the R –> PR transition. The finding that residues in the AWD also drive the kinetics of the R –> PR transition shows how the myosin architecture regulates the allosteric movements during the reaction cycle.

Understanding how monomeric proteins fold under in vitro conditions is crucial to describing their functions in the cellular context. Significant advances in theory and experiments have resulted in a conceptual framework for describing the folding mechanisms of globular proteins. The sizes of proteins in the denatured and folded states, cooperativity of the folding transition, dispersions in the melting temperatures at the residue level, and timescales of folding are, to a large extent, determined by N, the number of residues. The intricate details of folding as a function of denaturant concentration can be predicted by using a novel coarse-grained molecular transfer model. By watching one molecule fold at a time, using single-molecule methods, investigators have established the validity of the theoretically anticipated heterogeneity in the folding routes and the N-dependent timescales for the three stages in the approach to the native state. Despite the successes of theory, of which only a few examples are documented here, we conclude that much remains to be done to solve the protein folding problem in the broadest sense.

Using lattice models we explore the factors that determine the tendencies of polypeptide chains to aggregate by exhaustively sampling the sequence and conformational space. The morphologies of the fibril-like structures and the time scales (τ(fib)) for their formation depend on a balance between hydrophobic and Coulomb interactions. The extent of population of an ensemble of N* structures, which are fibril-prone structures in the spectrum of conformations of an isolated protein, is the major determinant of τ(fib). This observation is used to determine the aggregating sequences by exhaustively exploring the sequence space, thus providing a basis for genome wide search of fragments that are aggregation prone.

The interiors of cells are crowded, thus making it important to assess the effects of macromolecules on the folding of proteins. Using the self-organized polymer (SOP) model, which is a coarse-grained representation of polypeptide chains, we probe the mechanical stability of ubiquitin (Ub) monomers and trimers ((Ub)(3)) in the presence of monodisperse spherical crowding agents. Crowding increases the volume fraction (Phi(c))-dependent average force (f(u)(Phi(c))), relative to the value at Phi(c) = 0, needed to unfold Ub and the polyprotein. For a given Phi(c), the values of f(u)(Phi(c)) increase as the diameter (sigma(c)) of the crowding particles decreases. The average unfolding force f(u)(Phi(c)) depends on the ratio D/R(g), where D approximately sigma(c)(pi/6Phi(c))(1/3), with R(g) being the radius of gyration of Ub (or (Ub)(3)) in the unfolded state. Examination of the unfolding pathways shows that, relative to Phi(c) = 0, crowding promotes reassociation of ruptured secondary structural elements. Both the nature of the unfolding pathways and f(u)(Phi(c)) for (Ub)(3) are altered in the presence of crowding particles, with the effect being most dramatic for the subunit that unfolds last. We predict, based on SOP simulations and theoretical arguments, that f(u)(Phi(c)) approximately Phi(c)(1/3nu), where nu is the Flory exponent that describes the unfolded (random coil) state of the protein.

Determining the network of residues that transmit allosteric signals is crucial to understanding the function of biological nanomachines. During the course of a reaction cycle, biological machines in general, and Escherichia coli chaperonin GroEL in particular, undergo large-scale conformational changes in response to ligand binding. Normal mode analyses, based on structure-based coarse-grained models where each residue is represented by an alpha carbon atom, have been widely used to describe the motions encoded in the structures of proteins. Here, we propose a new Calpha-side chain elastic network model of proteins that includes information about the physical identity of each residue and accurately accounts for the side-chain topology and packing within the structure. Using the Calpha-side chain elastic network model and the structural perturbation method, which probes the response of a local perturbation at a given site at all other sites in the structure, we determine the network of key residues (allostery wiring diagram) responsible for the T–>R and R”–>T transitions in GroEL. A number of residues, both within a subunit and at the interface of two adjacent subunits, are found to be at the origin of the positive cooperativity in the ATP-driven T–>R transition. Of particular note are residues G244, R58, D83, E209, and K327. Of these, R38, D83, and K327 are highly conserved. G244 is located in the apical domain at the interface between two subunits; E209 and K327 are located in the apical domain, toward the center of a subunit; R58 and D83 are equatorial domain residues. The allostery wiring diagram shows that the network of residues are interspersed throughout the structure. Residues D83, V174, E191, and D359 play a critical role in the R”–>T transition, which implies that mutations of these residues would compromise the ATPase activity. D83 and E191 are also highly conserved; D359 is moderately conserved. The negative cooperativity between the rings in the R”–>T transition is orchestrated through several interface residues within a single ring, including N10, E434, D435, and E451. Signal from the trans ring that is transmitted across the interface between the equatorial domains is responsible for the R”–>T transition. The cochaperonin GroES plays a passive role in the R”–>T transition. Remarkably, the binding affinity of GroES for GroEL is allosterically linked to GroEL residues 350-365 that span helices K and L. The movements of helices K and L alter the polarity of the cavity throughout the GroEL functional cycle and undergo large-scale motions that are anticorrelated with the other apical domain residues. The allostery wiring diagrams for the T–>R and R”–>T transitions of GroEL provide a microscopic foundation for the cooperativity (anticooperativity) within (between) the ring (rings). Using statistical coupling analysis, we extract evolutionarily linked clusters of residues in GroEL and GroES. We find that several substrate protein binding residues as well as sites related to ATPase activity belong to a single functional network in GroEL. For GroES, the mobile loop residues and GroES/GroES interface residues are linked.

The coil-globule transition, a tenet of the physics of polymers, has been identified in recent years as an important unresolved aspect of the initial stages of the folding of proteins. We describe the basics of the collapse transition, starting with homopolymers and continuing with proteins. Studies of denatured-state collapse under equilibrium are then presented. An emphasis is placed on single-molecule fluorescence experiments, which are particularly useful for measuring properties of the denatured state even under conditions of coexistence with the folded state. Attempts to understand the dynamics of collapse, both theoretically and experimentally, are then described. Only an upper limit for the rate of collapse has been obtained so far. Improvements in experimental and theoretical methodology are likely to continue to push our understanding of the importance of the denatured-state thermodynamics and dynamics for protein folding in the coming years.

Structure-based elastic network models (ENMs) have been remarkably successful in describing conformational transitions in a variety of biological systems. Low-frequency normal modes are usually calculated from the ENM that characterizes elastic interactions between residues in contact in a given protein structure with a uniform force constant. To explore the dynamical effects of nonuniform elastic interactions, we calculate the robustness and coupling of the low-frequency modes in the presence of nonuniform variations in the ENM force constant. The variations in the elastic interactions, approximated here by Gaussian noise, approximately account for perturbation effects of heterogeneous residue-residue interactions or evolutionary sequence changes within a protein family. First-order perturbation theory provides an efficient and qualitatively correct estimate of the mode robustness and mode coupling for finite perturbations to the ENM force constant. The mode coupling analysis and the mode robustness analysis identify groups of strongly coupled modes that encode for protein functional motions. We illustrate the new concepts using myosin II motor protein as an example. The biological implications of mode coupling in tuning the allosteric couplings among the actin-binding site, the nucleotide-binding site, and the force-generating converter and lever arm in myosin isoforms are discussed. We evaluate the robustness of the correlation functions that quantify the allosteric couplings among these three key structural motifs.

Recent experiments have shown that the congener Abeta(1-40)[D23-K28], in which the side chains of charged residues Asp23 and Lys28 are linked by a lactam bridge, forms amyloid fibrils that are structurally similar to the wild type (WT) Abeta peptide, but at a rate that is nearly 1000 times faster. We used all atom molecular dynamics simulations in explicit water, and two force fields, of the WT dimer, a monomer with the lactam bridge (Abeta(10-35)-lactam[D23-K28]), and the monomer and dimers with harmonically constrained D23-K28 salt bridge (Abeta(10-35)[D23-K28]) to understand the origin of the enhanced fibril rate formation. The simulations show that the assembly competent fibril-like monomer (N*) structure, which is present among the conformations sampled by the isolated monomer, with strand conformations in the residues spanning the N and C termini and a bend involving residues D(23) VGSNKG(29), are populated to a much greater extent in Abeta(10-35)[D23-K28] and Abeta(10-35)-lactam[D23-K28] than in the WT, which has negligible probability of forming N*. The salt bridge in N* of Abeta(10-35)[D23-K28], whose topology is similar to that found in the fibril, is hydrated. The reduction in the free energy barrier to fibril formation in Abeta(10-35)[D23-K28] and in Abeta(10-35)-lactam[D23-K28], compared to the WT, arises largely due to entropic restriction which enables the bend formation. A decrease in the entropy of the unfolded state and the lesser penalty for conformational rearrangement including the formation of the salt bridge in Abeta peptides with D23-K28 constraint results in a reduction in the kinetic barrier in the Abeta(1-40)-lactam[D23-K28] congener compared to the WT. The decrease in the barrier, which is related to the free energy cost of forming a bend, is estimated to be in the range (4-7)k(B)T. Although a number of factors determine the growth of fibrils, the decrease in the free energy barrier, relative to the WT, to N* formation is a major factor in the rate enhancement in the fibril formation of Abeta(1-40)[D23-K28] congener. Qualitatively similar results were obtained using simulations of Abeta(9-40) peptides and various constructs related to the Abeta(10-35) systems that were probed using OPLS and CHARMM force fields. We hypothesize that mutations or other constraints that preferentially enhance the population of the N* species would speed up aggregation rates. Conversely, ligands that lock it in the fibril-like N* structure would prevent amyloid formation.

Experiments show that for many two-state folders the free energy of the native state, DeltaG(ND)([C]), changes linearly as the denaturant concentration, [C], is varied. The slope {m = [dDeltaG(ND)([C])]/(d[C])}, is nearly constant. According to the transfer model, the m-value is associated with the difference in the surface area between the native (N) and denatured (D) state, which should be a function of DeltaR(g)(2), the difference in the square of the radius of gyration between the D and N states. Single-molecule experiments show that the R(g) of the structurally heterogeneous denatured state undergoes an equilibrium collapse transition as [C] decreases, which implies m also should be [C]-dependent. We resolve the conundrum between constant m-values and [C]-dependent changes in R(g) using molecular simulations of a coarse-grained representation of protein L, and the molecular transfer model, for which the equilibrium folding can be accurately calculated as a function of denaturant (urea) concentration. In agreement with experiment, we find that over a large range of denaturant concentration (>3 M) the m-value is a constant, whereas under strongly renaturing conditions (<3 M), it depends on [C]. The m-value is a constant above [C] > 3 M because the [C]-dependent changes in the surface area of the backbone groups, which make the largest contribution to m, are relatively narrow in the denatured state. The burial of the backbone and hydrophobic side chains gives rise to substantial surface area changes below [C] < 3 M, leading to collapse in the denatured state of protein L. Dissection of the contribution of various amino acids to the total surface area change with [C] shows that both the sequence context and residual structure are important. There are [C]-dependent variations in the surface area for chemically identical groups such as the backbone or Ala. Consequently, the midpoints of transition of individual residues vary significantly (which we call the Holtzer effect) even though global folding can be described as an all-or-none transition. The collapse is specific in nature, resulting in the formation of compact structures with appreciable populations of nativelike secondary structural elements. The collapse transition is driven by the loss of favorable residue-solvent interactions and a concomitant increase in the strength of intrapeptide interactions with a decreasing [C]. The strength of these interactions is nonuniformly distributed throughout the structure of protein L. Certain secondary structure elements have stronger [C]-dependent interactions than others in the denatured state.

We develop an analytical method for studying the properties of a noninteracting wormlike chain (WLC) in confined geometries. The mean-field-like theory replaces the rigid constraints of confinement with average constraints, thus allowing us to develop a tractable method for treating a WLC wrapped on the surface of a sphere, and fully encapsulated within it. The efficacy of the theory is established by reproducing the exact correlation functions for a WLC confined to the surface of a sphere. In addition, the coefficients in the free energy are exactly calculated. We also describe the behavior of a surface-confined chain under external tension that is relevant for single molecule experiments on histone-DNA complexes. The force-extension curves display spatial oscillations, and the extension of the chain, whose maximum value is bounded by the sphere diameter, scales as f(-1) at large forces, in contrast to the unconfined chain that approaches the contour length as f(-1/2). A WLC encapsulated in a sphere, that is relevant for the study of the viral encapsulation of DNA, can also be treated using the mean-field approach. The predictions of the theory for various correlation functions are in excellent agreement with Langevin simulations. We find that strongly confined chains are highly structured by examining the correlations using a local winding axis. The predicted pressure of the system is in excellent agreement with simulations but, as is known, is significantly lower than the pressures seen for DNA packaged in viral capsids.

Abeta peptide is an essential protein in the pathogenesis of Alzheimer’s disease and is derived from amyloid precursor protein (APP) in the membrane by beta- and gamma-secretase cleavage. An experimental study has shown that a pairwise replacement of Gly with Leu in APP enhances homodimerization but leads to a drastic reduction of Abeta secretion. To resolve this apparent discrepancy, we predicted the wild-type (WT) and mutant APP dimer conformations by replica-exchange molecular dynamics simulations using the implicit membrane model IMM1. The simulations illustrate large conformational differences between the WT and mutant APP fragments in the membrane. Dimerization of the WT is due to two C(alpha)-H…O hydrogen bonds between two APP fragments, whereas dimerization of the mutant is due to hydrophobic interactions. In the mutant, each APP fragment is more tilted, and the gamma-cleavage site is shifted toward the center of the membrane. This position produces a mismatch between the active site of gamma-secretase and the gamma-cleavage site of APP that might prohibit Abeta production.

We use Metropolis Monte Carlo and umbrella sampling to calculate the free energies of interaction of two methane molecules and their charged derivatives in cylindrical water-filled pores. Confinement strongly alters the interactions between the nonpolar solutes and completely eliminates the solvent separated minimum (SSM) that is seen in bulk water. The free energy profiles show that the methane molecules are either in contact or at separations corresponding to the diameter and the length of the cylindrical pore. Analytic calculations that estimate the entropy of the solutes, which are solvated at the pore surface, qualitatively explain the shape of the free energy profiles. Adding charges of opposite sign and magnitude 0.4e or e (where e is the electronic charge) to the methane molecules decreases their tendency for surface solvation and restores the SSM. We show that confinement induced ion-pair formation occurs whenever l(B)/D approximately O(1), where l(B) is the Bjerrum length and D is the pore diameter. The extent of stabilization of the SSM increases with ion charge density as long as l(B)/D<1. In pores with D

It is suspected that correlated motions among a subset of spatially separated residues drive conformational dynamics not only in multidomain but also in single domain proteins. Sequence and structure-based methods have been proposed to determine covariation between two sites on a protein. The statistical coupling analysis (SCA) that compares the changes in probability at two sites in a multiple sequence alignment (MSA) and a subset of the MSA has been used to infer the network of residues that encodes allosteric signals in protein families. The structural perturbation method (SPM), that probes the response of a local perturbation at all other sites, has been used to probe the allostery wiring diagram in biological machines and enzymes. To assess the efficacy of the SCA, we used an exactly soluble two dimensional lattice model and performed double-mutant cycle (DMC) calculations to predict the extent of physical coupling between two sites. The predictions of the SCA and the DMC results show that only residues that are in contact in the native state are accurately identified. In addition, covariations among strongly interacting residues are most easily identified by the SCA. These conclusions are consistent with the DMC experiments on the PDZ family. Good correlation between the SCA and the DMC is only obtained by performing multiple experiments that vary the nature of amino acids at a given site. In contrast, the energetic coupling found in experiments for the PDZ domain are recovered using the SPM. We also predict, using the SPM, several residues that are coupled energetically.

Allostery forms the basis of intra-molecular communications in various enzymes, however the underlying conformational changes are largely elusive. Recently, we have proposed to employ an elastic model based normal mode analysis to investigate the allosteric transitions in several molecular nanomachines (including myosin II, DNA polymerase and chaperonin GroEL). After combining with bioinformatics analysis of the evolutionary sequence variations, we have been able to identify the highly conserved and robust modes of collective motions that are capable of transmitting molecular signals over long distances.

Understanding how RNA molecules navigate their rugged folding landscapes holds the key to describing their roles in a variety of cellular functions. To dissect RNA folding at the molecular level, we performed simulations of three pseudoknots (MMTV and SRV-1 from viral genomes and the hTR pseudoknot from human telomerase) using coarse-grained models. The melting temperatures from the specific heat profiles are in good agreement with the available experimental data for MMTV and hTR. The equilibrium free energy profiles, which predict the structural transitions that occur at each melting temperature, are used to propose that the relative stabilities of the isolated helices control their folding mechanisms. Kinetic simulations, which corroborate the inferences drawn from the free energy profiles, show that MMTV folds by a hierarchical mechanism with parallel paths, i.e., formation of one of the helices nucleates the assembly of the rest of the structure. The SRV-1 pseudoknot, which folds in a highly cooperative manner, assembles in a single step in which the preformed helices coalesce nearly simultaneously to form the tertiary structure. Folding occurs by multiple pathways in the hTR pseudoknot, the isolated structural elements of which have similar stabilities. In one of the paths, tertiary interactions are established before the formation of the secondary structures. Our work shows that there are significant sequence-dependent variations in the folding landscapes of RNA molecules with similar fold. We also establish that assembly mechanisms can be predicted using the stabilities of the isolated secondary structures.

We elucidate the structural transitions in a helical off-lattice homopolymer induced by crowding agents, as a function of the number of monomers (N) and volume fraction (varphi c) of crowding particles. At varphic=0, the homopolymer undergoes transitions from a random coil to a helix, helical hairpin HH, and helix bundle HB structures depending on N, and temperature. Crowding induces chain compaction that can promote HH or HB formation depending on varphic. Typically, the helical content decreases which is reflected in the decrease in the transition temperatures that depend on varphic, N, and the size of the crowding particles.

Sequence-dependent variations in the growth mechanism and stability of amyloid fibrils, which are implicated in a number of neurodegenerative diseases, are poorly understood. We have carried out extensive all-atom molecular dynamics simulations to monitor the structural changes that occur upon addition of random coil (RC) monomer fragments from the yeast prion Sup35 and Abeta-peptide onto a preformed fibril. Using the atomic resolution structures of the microcrystals as the starting points, we show that the RC –> beta-strand transition for the Sup35 fragment occurs abruptly over a very narrow time interval, whereas the acquisition of strand content is less dramatic for the hydrophobic-rich Abeta-peptide. Expulsion of water, resulting in the formation of a dry interface between 2 adjacent sheets of the Sup35 fibril, occurs in 2 stages. Ejection of a small number of discrete water molecules in the second stage follows a rapid decrease in the number of water molecules in the first stage. Stability of the Sup35 fibril is increased by a network of hydrogen bonds involving both backbone and side chains, whereas the marginal stability of the Abeta-fibrils is largely due to the formation of weak dispersion interaction between the hydrophobic side chains. The importance of the network of hydrogen bonds is further illustrated by mutational studies, which show that substitution of the Asn and Gln residues to Ala compromises the Sup35 fibril stability. Despite the similarity in the architecture of the amyloid fibrils, the growth mechanism and stability of the fibrils depend dramatically on the sequence.

The influence of hydration on the nanosecond timescale dynamics of tRNA is investigated using neutron scattering spectroscopy. Unlike protein dynamics, the dynamics of tRNA is not affected by methyl group rotation. This allows for a simpler analysis of the influence of hydration on the conformational motions in RNA. We find that hydration affects the dynamics of tRNA significantly more than that of lysozyme. Both the characteristic length scale and the timescale of the conformational motions in tRNA depend strongly on hydration. Even the characteristic temperature of the so-called “dynamical transition” appears to be hydration-dependent in tRNA. The amplitude of the conformational motions in fully hydrated tRNA is almost twice as large as in hydrated lysozyme. We ascribe these differences to a more open and flexible structure of hydrated RNA, and to a larger fraction and different nature of hydrophilic sites. The latter leads to a higher density of water that makes the biomolecule more flexible. All-atom molecular-dynamics simulations are used to show that the extent of hydration is greater in tRNA than in lysozyme. We propose that water acts as a “lubricant” in facilitating enhanced motion in solvated RNA molecules.

Positively charged counterions drive RNA molecules into compact configurations that lead to their biologically active structures. To understand how the valence and size of the cations influences the collapse transition in RNA, small-angle X-ray scattering was used to follow the decrease in the radius of gyration (R(g)) of the Azoarcus and Tetrahymena ribozymes in different cations. Small, multivalent cations induced the collapse of both ribozymes more efficiently than did monovalent ions. Thus, the cooperativity of the collapse transition depends on the counterion charge density. Singular value decomposition of the scattering curves showed that folding of the smaller and more thermostable Azoarcus ribozyme is well described by two components, whereas collapse of the larger Tetrahymena ribozyme involves at least one intermediate. The ion-dependent persistence length, extracted from the distance distribution of the scattering vectors, shows that the Azoarcus ribozyme is less flexible at the midpoint of transition in low-charge-density ions than in high-charge-density ions. We conclude that the formation of sequence-specific tertiary interactions in the Azoarcus ribozyme overlaps with neutralization of the phosphate charge, while tertiary folding of the Tetrahymena ribozyme requires additional counterions. Thus, the stability of the RNA structure determines its sensitivity to the valence and size of the counterions.

Fibril formation from amyloidogenic peptides is a hallmark of a wide range of diseases, including Alzheimer’s disease and type II diabetes. Characterization of the aggregation process should include intrinsic factors, such as sequence variation, and extrinsic factors, such as crowding effects. To this end, we examined the interactions of dimers composed of residues 20-29 of human islet amyloid polypeptide (hIAPP), which form fibrils in vitro, and the nonamyloidogenic rat IAPP (rIAPP) using molecular dynamics simulations modeled at different peptide concentrations. There is a substantial free energy barrier to unbind the hIAPP dimer whereas no barrier exists for separating the rIAPP dimer. The profound differences in the free energy landscapes of the rIAPP and hIAPP dimers explains the lack of fibril formation in hIAPP upon substitution of the C-terminal residues by proline. Enhancing the extent of crowding has a substantial effect on both the barrier for separating a hIAPP beta-sheet dimer and the formation of potential beta-sheet nucleation sites. Our results show that the propensity for forming nucleation sites is dependent not only on the amino-acid sequence but also on the context in which it is found.

Single-molecule force spectroscopy methods can be used to generate folding trajectories of biopolymers from arbitrary regions of the folding landscape. We illustrate the complexity of the folding kinetics and generic aspects of the collapse of RNA and proteins upon force quench by using simulations of an RNA hairpin and theory based on the de Gennes model for homopolymer collapse. The folding time, tau(F), depends asymmetrically on deltaf(S) = f (S) – f (m) and deltaf (Q) = f (m) – f (Q) where f (S) (f (Q)) is the stretch (quench) force and f (m) is the transition midforce of the RNA hairpin. In accord with experiments, the relaxation kinetics of the molecular extension, R(t), occurs in three stages: A rapid initial decrease in the extension is followed by a plateau and finally, an abrupt reduction in R(t) occurs as the native state is approached. The duration of the plateau increases as lambda = tau (Q)/tau (F) decreases (where tau (Q) is the time in which the force is reduced from f (S) to f (Q)). Variations in the mechanisms of force-quench relaxation as lambda is altered are reflected in the experimentally measurable time-dependent entropy, which is computed directly from the folding trajectories. An analytical solution of the de Gennes model under tension reproduces the multistage stage kinetics in R(t). The prediction that the initial stages of collapse should also be a generic feature of polymers is validated by simulation of the kinetics of toroid (globule) formation in semiflexible (flexible) homopolymers in poor solvents upon quenching the force from a fully stretched state. Our findings give a unified explanation for multiple disparate experimental observations of protein folding.

Aggregation of Amyloid beta (Abeta) peptide has been linked to the neurodegenerative Alzheimer’s Disease and implicated in other amyloid diseases including cerebral amyloid angiopathy. Abeta peptide is generated by cleavage of the amyloid precursor protein (APP) by transmembrane proteases. It is crucial to determine the structures of beta-amyloid peptides in a membrane to provide a molecular basis for the cleavage mechanism. We report the structures of amyloid beta peptide (Abeta(1-40) and Abeta(1-42)) as well as the 672-726 fragment of APP (referred to as Abeta(1-55)) in a membrane environment determined by replica-exchange molecular dynamics simulation. Abeta(1-40) is found to have two helical domains A (13-22) and B(30-35) and a type I beta-turn at 23-27. The peptide is localized at the interface between membrane and solvent. Substantial fluctuations in domain A are observed. The dominant simulated tertiary structure of Abeta(1-40) is observed to be similar to the simulated Abeta(1-42) structure. However, there are differences observed in the overall conformational ensemble, as characterized by the two-dimensional free energy surfaces. The fragment of APP (Abeta(1-55)) is observed to have a long transmembrane helix. The position of the transmembrane region and ensemble of membrane structures are elucidated. The conformational transition between the transmembrane Abeta(1-55) structure, prior to cleavage, and the Abeta(1-40) structure, following cleavage, is proposed.

The mechanism of addition of a soluble unstructured monomer to a preformed ordered amyloid fibril is a complex process. On the basis of the kinetics of monomer disassociation of Abeta(1-40) from the amyloid fibril, it has been suggested that deposition is a multistep process involving a rapid reversible association of the unstructured monomer to the fibril surface (docking) followed by a slower conformational rearrangement leading to the incorporation onto the underlying fibril lattice (locking). By exploiting the vast time scale separation between the dock and lock processes and using molecular dynamics simulation of deposition of the disordered peptide fragment (35)MVGGVV(40) from the Abeta peptide onto the fibril with known crystal structure, we provide a thermodynamic basis for the dock-lock mechanism of fibril growth. Free energy profiles, computed using implicit solvent model and enhanced sampling methods with the distance (delta(C)) between the center of mass of the peptide and the fibril surface as the order parameter, show three distinct basins of attraction. When delta(C) is large, the monomer is compact and unstructured and the favorable interactions with the fibril results in stretching of the peptide at delta(C) approximately 13 A. As delta(C) is further decreased, the peptide docks onto the fibril surface with a structure that is determined by a balance between intrapeptide and peptide fibril interactions. At delta(C) approximately 4 A, a value that is commensurate with the spacing between beta-strands in the fibril, the monomer expands and locks onto the fibril. Using simulations with implicit solvent model and all atom molecular dynamics in explicit water, we show that the locked monomer, which interacts with the underlying fibril, undergoes substantial conformational fluctuations and is not stable. The cosolutes urea and TMAO destabilize the unbound phase and stabilize the docked phase. Interestingly, small crowding particles enhance the stability of the fibril-bound monomer only marginally. We predict that the experimentally measurable critical monomer concentration, C(R), at which the soluble unbound monomer is in equilibrium with the ordered fibril, increases sharply as temperature is increased under all solution conditions.

Urea titration of RNA by urea is an effective approach to investigate the forces stabilizing this biologically important molecule. We used all atom molecular dynamics simulations using two urea force fields and two RNA constructs to elucidate in atomic detail the destabilization mechanism of folded RNA in aqueous urea solutions. Urea denatures RNA by forming multiple hydrogen bonds with the RNA bases and has little influence on the phosphodiester backbone. Most significantly we discovered that urea engages in stacking interactions with the bases. We also estimate, for the first time, the m-value for RNA, which is a measure of the strength of urea-RNA interactions. Our work provides a conceptual understanding of the mechanism by which urea enhances RNA folding rates.

Knowledge-based contact potentials are routinely used in fold recognition, binding of peptides to proteins, structure prediction, and coarse-grained models to probe protein folding kinetics. The dominant physical forces embodied in the contact potentials are revealed by eigenvalue analysis of the matrices, whose elements describe the strengths of interaction between amino acid side chains. We propose a general method to rank quantitatively the importance of various inter-residue interactions represented in the currently popular pair contact potentials. Eigenvalue analysis and correlation diagrams are used to rank the inter-residue pair interactions with respect to the magnitude of their relative contributions to the contact potentials. The amino acid ranking is shown to be consistent with a mean field approximation that is used to reconstruct the original contact potentials from the most relevant amino acids for several contact potentials. By providing a general, relative ranking score for amino acids, this method permits a detailed, quantitative comparison of various contact interaction schemes. For most contact potentials, between 7 and 9 amino acids of varying chemical character are needed to accurately reconstruct the full matrix. By correlating the identified important amino acid residues in contact potentials and analysis of about 7800 structural domains in the CATH database we predict that it is important to model accurately interactions between small hydrophobic residues. In addition, only potentials that take interactions involving the protein backbone into account can predict dense packing in protein structures.

We study the effect of the osmolyte, Trimethylamine N-Oxide (TMAO), which accumulates in cells in response to osmotic stress, on the stability of RNA hairpins. All atom molecular dynamics (MD) simulations of a nucleotide and the 22-nucleotide RNA hairpin P5GA in an aqueous TMAO solution show that TMAO preferentially interacts with the base through the formation of a single hydrogen bond. To circumvent the difficulties of adequately sampling the conformational space of polynucleotides, we used coarse-grained models (including one that is inspired by the results of all-atom MD simulations of a single nucleotide) to probe the effects of osmoyltes on the stability of P5GA. If, as revealed by our MD simulations, the cosolute specifically interacts with only one base at a time, then we find practically no change in hairpin stability as measured by Delta T m = T m(Phi) – T m, where T m(Phi) and T m are the melting temperatures at volume fraction Phi of the osmolyte and Phi = 0, respectively. This finding is in qualitative agreement with recent experiments. If the interactions between the RNA and osmolytes are repulsive, which is appropriate for mimicking the effects of crowding, Delta T m can vary from 5 to 15 K depending on the size of the osmolyte and the nature of RNA-osmolyte interactions. Cosolutes that interact favorably with multiple bases simultaneously can stabilize the hairpin more than a crowding agent of the same size. The implications of our predictions for experiments are briefly outlined.

The bacterial chaperonin GroEL and the co-chaperonin GroES assist in the folding of a number of structurally unrelated substrate proteins (SPs). In the absence of chaperonins, SP folds by the kinetic partitioning mechanism (KPM), according to which a fraction of unfolded molecules reaches the native state directly, while the remaining fraction gets trapped in a potentially aggregation-prone misfolded state. During the catalytic reaction cycle, GroEL undergoes a series of allosteric transitions (T<–>R–>R”–>T) triggered by SP capture, ATP binding and hydrolysis, and GroES binding. We developed a general kinetic model that takes into account the coupling between the rates of the allosteric transitions and the folding and aggregation of the SP. Our model, in which the GroEL allosteric rates and SP-dependent folding and aggregation rates are independently varied without prior assumption, quantitatively fits the GroEL concentration-dependent data on the yield of native ribulose bisphosphate carboxylase/oxygenase (Rubisco) as a function of time. The extracted kinetic parameters for the GroEL reaction cycle are consistent with the available values from independent experiments. In addition, we also obtained physically reasonable parameters for the kinetic steps in the reaction cycle that are difficult to measure. If experimental values for GroEL allosteric rates are used, the time-dependent changes in native-state yield at eight GroEL concentrations can be quantitatively fit using only three SP-dependent parameters. The model predicts that the differences in the efficiencies (as measured by yields of the native state) of GroEL, single-ring mutant (SR1), and variants of SR1, in the rescue of mitochondrial malate dehydrogenase, citrate synthase, and Rubisco, are related to the large variations in the allosteric transition rates. We also show that GroEL/S mutants that efficiently fold one SP at the expense of all others are due to a decrease in the rate of a key step in the reaction cycle, which implies that wild-type GroEL has evolved as a compromise between generality and specificity. We predict that, under maximum loading conditions and saturating ATP concentration, the efficiency of GroEL (using parameters for Rubisco) depends predominantly on the rate of R–>R” transition, while the equilibrium constant of the T<–>R has a small effect only. Both under sub- and superstoichiometric GroEL concentrations, enhanced efficiency is achieved by rapid turnover of the reaction cycle, which is in accord with the predictions of the iterative annealing mechanism. The effects are most dramatic at substoichiometric conditions (most relevant for in vivo situations) when SP aggregation can outcompete capture of SP by chaperonins.

We investigate the kinetics of loop formation in ideal flexible polymer chains (the Rouse model), and polymers in good and poor solvents. We show for the Rouse model, using a modification of the theory of Szabo, Schulten, and Schulten, that the time scale for cyclization is tau(c) approximately tau(0)N(2) (where tau(0) is a microscopic time scale and N is the number of monomers), provided the coupling between the relaxation dynamics of the end-to-end vector and the looping dynamics is taken into account. The resulting analytic expression fits the simulation results accurately when a, the capture radius for contact formation, exceeds b, the average distance between two connected beads. Simulations also show that when a < b, tau(c) approximately N(alpha)(tau), where 1.5 < alpha(tau) < or = 2 in the range 7 < N < 200 used in the simulations. By using a diffusion coefficient that is dependent on the length scales a and b (with a < b), which captures the two-stage mechanism by which looping occurs when a < b, we obtain an analytic expression for tauc that fits the simulation results well. The kinetics of contact formation between the ends of the chain are profoundly effected when interactions between monomers are taken into account. Remarkably, for N < 100, the values of tau(c) decrease by more than 2 orders of magnitude when the solvent quality changes from good to poor. Fits of the simulation data for tau(c) to a power law in N (tau(c) approximately N(alpha)(tau)) show that alpha(tau) varies from about 2.4 in a good solvent to about 1.0 in poor solvents. The effective exponent alpha(tau) decreases as the strength of the attractive monomer-monomer interactions increases. Loop formation in poor solvents, in which the polymer adopts dense, compact globular conformations, occurs by a reptation-like mechanism of the ends of the chain. The time for contact formation between beads that are interior to the chain in good solvents changes nonmonotonically as the loop length varies. In contrast, the variation in interior loop closure time is monotonic in poor solvents. The implications of our results for contact formation in polypeptide chains, RNA, and single-stranded DNA are briefly outlined.

The initial events in protein aggregation involve fluctuations that populate monomer conformations, which lead to oligomerization and fibril assembly. The highly populated structures, driven by a balance between hydrophobic and electrostatic interactions in the protease-resistant wild-type Abeta(21-30) peptide and mutants E22Q (Dutch), D23N (Iowa), and K28N, are analyzed using molecular dynamics simulations. Intrapeptide electrostatic interactions were connected to calculated pK(a) values that compare well with the experimental estimates. The pK(a) values of the titratable residues show that E22 and D23 side chains form salt bridges only infrequently with the K28 side chain. Contacts between E22-K28 are more probable in “dried” salt bridges, whereas D23-K28 contacts are more probable in solvated salt bridges. The strength of the intrapeptide hydrophobic interactions increases as D23N

Force-induced deformations of a self-avoiding chain confined inside a cylindrical cavity, with diameter D, are probed using molecular dynamics simulations, scaling analysis, and analytical calculations. We obtain and confirm a simple scaling relation -fD approximately R(-9/4) in the strong-compression regime, while for weak deformations, we find fD = -A(R/R0) + B(R/R0)(-2), where A and B are constants, f the external force, and R the chain extension (with R0 its unperturbed value). For a strong stretch, we present a universal, analytical force-extension relation. Our results can be used to analyze the behavior of biomolecules in confinement.

Interactions between denaturants and proteins are commonly used to probe the structures of the denatured state ensemble and their stabilities. Osmolytes, a class of small intracellular organic molecules found in all taxa, also profoundly affect the equilibrium properties of proteins. We introduce the molecular transfer model, which combines simulations in the absence of denaturants or osmolytes, and Tanford’s transfer model to predict the dependence of equilibrium properties of proteins at finite concentration of osmolytes. The calculated changes in the thermodynamic quantities (probability of being in the native basin of attraction, m values, FRET efficiency, and structures of the denatured state ensemble) with GdmCl concentration [C] for the protein L and cold shock protein CspTm compare well with experiments. The radii of gyration of the subpopulation of unfolded molecules for both proteins decrease (i.e., they undergo a collapse transition) as [C] decreases. Although global folding is cooperative, residual secondary structures persist at high denaturant concentrations. The temperature dependence of the specific heat shows that the folding temperature (T(F)) changes linearly as urea and trimethylamine N-oxide (TMAO) concentrations increase. The increase in T(F) in TMAO can be as large as 20 degrees C, whereas urea decreases T(F) by as much as 35 degrees C. The stabilities of protein L and CspTm also increase linearly with the concentration of osmolytes (proline, sorbitol, sucrose, TMAO, and sarcosine).

The effect of confinement on the stability and dynamics of peptides and proteins is relevant in the context of a number of problems in biology and biotechnology. We have examined the stability of different helix-forming sequences upon confinement to a carbon nanotube using Langevin dynamics simulations of a coarse-grained representation of the polypeptide chain. We show that the interplay of several factors that include sequence, solvent conditions, strength (lambda) of nanotube-peptide interactions, and the nanotube diameter (D) determines confinement-induced stability of helicies. In agreement with predictions based on polymer theory, the helical state is entropically stabilized for all sequences when the interaction between the peptide and the nanotube is weakly hydrophobic and D is small. However, there is a strong sequence dependence as the strength of the lambda increases. For an amphiphilic sequence, the helical stability increases with lambda, whereas for polyalanine the diagram of states is a complex function of lambda and D. In addition, decreasing the size of the “hydrophobic patch” lining the nanotube, which mimics the chemical heterogeneity of the ribosome tunnel, increases the helical stability of the polyalanine sequence. Our results provide a framework for interpreting a number of experiments involving the structure formation of peptides in the ribosome tunnel as well as transport of biopolymers through nanotubes.

The sequence-dependent folding landscapes of nucleic acid hairpins reflect much of the complexity of biomolecular folding. Folding trajectories, generated by using single-molecule force-clamp experiments by attaching semiflexible polymers to the ends of hairpins, have been used to infer their folding landscapes. Using simulations and theory, we study the effect of the dynamics of the attached handles on the handle-free RNA free-energy profile F(o)(eq)(z(m)), where z(m) is the molecular extension of the hairpin. Accurate measurements of F(o)(eq)(z(m)) requires stiff polymers with small L/l(p), where L is the contour length of the handle, and l(p) is the persistence length. Paradoxically, reliable estimates of the hopping rates can only be made by using flexible handles. Nevertheless, we show that the equilibrium free-energy profile F(o)(eq)(z(m)) at an external tension f(m), the force (f) at which the folded and unfolded states are equally populated, in conjunction with Kramers’ theory, can provide accurate estimates of the force-dependent hopping rates in the absence of handles at arbitrary values of f. Our theoretical framework shows that z(m) is a good reaction coordinate for nucleic acid hairpins under tension.

Confinement effects on protein stability are relevant in a number of biological applications ranging from encapsulation in the cylindrical cavity of a chaperonin, translocation through pores, and structure formation in the exit tunnel of the ribosome. Consequently, free energies of interaction between amino acid side chains in restricted spaces can provide insights into factors that control protein stability in nanopores. Using all-atom molecular dynamics simulations, we show that 3 pair interactions between side chains–hydrophobic (Ala-Phe), polar (Ser-Asn) and charged (Lys-Glu)–are substantially altered in hydrophobic, water-filled nanopores, relative to bulk water. When the pore holds water at bulk density, the hydrophobic pair is strongly destabilized and is driven to large separations corresponding to the width and the length of the cylindrical pore. As the water density is reduced, the preference of Ala and Phe to be at the boundary decreases, and the contact pair is preferred. A model that accounts for the volume accessible to Phe and Ala in the solvent-depleted region near the pore boundary explains the simulation results. In the pore, the hydrogen-bonded interactions between Ser and Asn have an enhanced dependence on their relative orientations, as compared with bulk water. When the side chains of Lys and Glu are restrained to be side by side, parallel to each other, then salt bridge formation is promoted in the nanopore. Based on these results, we argue and demonstrate that for a generic amphiphilic sequence, cylindrical confinement is likely to enhance thermodynamic stability relative to the bulk.

Using exhaustive Monte Carlo simulations we study the kinetics and mechanism of fibril formation using lattice models as a function of temperature (T) and the number of chains (M). While these models are, at best, caricatures of peptides, we show that a number of generic features thought to govern fibril assembly are captured by the toy model. The monomer, which contains eight beads made from three letters (hydrophobic, polar, and charged), adopts a compact conformation in the native state. In both the single-layered protofilament (seen for M10) structures, the monomers are arranged in an antiparallel fashion with the “strandlike” conformation that is perpendicular to the fibril axis. Partial unfolding of the folded monomer that populates an aggregation prone conformation (N(*)) is required for ordered assembly. The contacts in the N(*) conformation, which is one of the four structures in the first “excited” state of the monomer, are also present in the native conformation. The time scale for fibril formation is a minimum in the T-range when the conformation N(*) is substantially populated. The kinetics of fibril assembly occurs in three distinct stages. In each stage there is a cascade of events that transforms the monomers and oligomers to ordered structures. In the first “burst” stage, highly mobile oligomers of varying sizes form. The conversion to the N(*) conformation occurs within the oligomers during the second stage in which a vast number of interchain contacts are established. As time progresses, a dominant cluster emerges that contains a majority of the chains. In the final stage, the aggregation of N(*) particles serve as a template onto which smaller oligomers or monomers can dock and undergo conversion to fibril structures. The overall time for growth in the latter stages is well described by the Lifshitz-Slyazov growth kinetics for crystallization from supersaturated solutions. The detailed analysis shows that elements of the three popular models, namely, nucleation and growth, templated assembly, and nucleated conformational conversion are present at various stages of fibril assembly.

Riboswitches, whose folding is controlled by binding of metabolites to the aptamer domain, regulate downstream gene expression. Folding properties of the aptamer strongly influence the conformation of the downstream expression platform, which controls transcription termination or translation initiation. We have characterized the energy landscape of the add riboswitch aptamer quantitatively by unfolding and refolding the molecule with mechanical force using the coarse-grained self-organized polymer model and Brownian dynamics simulation. Multiple folding states have been found during the folding process of the aptamer, both with and without adenine, consistent with single molecule studies of purine riboswitches. Adenine binding stabilizes the folded structure and significantly decreases the unfolding rate of the aptamer, the folding of which is in competition with the formation of the downstream stem-loop structure in the complete riboswitch. These results provide insights into the mechanism of gene regulation by the RNA switches.

We study the forced rupture of adhesive contacts between monomers that are not covalently linked in a Rouse chain. When the applied force (f) to the chain end is less than the critical force for rupture (f{c}), the reversible rupture process is coupled to the internal Rouse modes. If f/f{c}>1 the rupture is irreversible. In both limits, the nonexponential distribution of contact lifetimes, which depends sensitively on the location of the contact, follows the double-exponential (Gumbel) distribution. When two contacts are well separated along the chain, the rate limiting step in the sequential rupture kinetics is the disruption of the contact that is in the chain interior. If the two contacts are close to each other, they cooperate to sustain the stress, which results in an “all-or-none” transition.

The mechanism of denaturation of proteins by urea is explored by using all-atom microseconds molecular dynamics simulations of hen lysozyme generated on BlueGene/L. Accumulation of urea around lysozyme shows that water molecules are expelled from the first hydration shell of the protein. We observe a 2-stage penetration of the protein, with urea penetrating the hydrophobic core before water, forming a “dry globule.” The direct dispersion interaction between urea and the protein backbone and side chains is stronger than for water, which gives rise to the intrusion of urea into the protein interior and to urea’s preferential binding to all regions of the protein. This is augmented by preferential hydrogen bond formation between the urea carbonyl and the backbone amides that contributes to the breaking of intrabackbone hydrogen bonds. Our study supports the “direct interaction mechanism” whereby urea has a stronger dispersion interaction with protein than water.

Folding of many single-domain proteins has been described using the nucleation-collapse (NC) mechanism. According to NC, folding (formation of secondary structures and tertiary interactions) and chain collapse occur synchronously upon formation of native-like structures involving a critical number of residues. Using simple nucleation theory together with structure-based thermodynamic data, the average size of the most probable nucleus N(R)*, for single-domain proteins, is estimated to be between 15 and 30 residues. We argue that finite-sized fluctuations in this estimate can be large so that nearly half of the residues of a 100 residue protein can be part of the folding nucleus. Inclusion of surface area changes in the folded and unfolded states are important in the determination of N(R)*.

The complex role played by intermediates is dissected using experimental data on apomyoglobin (apoMb), simple theoretical concepts, and simulations of kinetics of simple minimal off-lattice models. The folding of moderate-to-large-sized proteins often occurs through passage of an ensemble of intermediates. In the case of apoMb there is dominant kinetic intermediate I that also occurs at equilibrium. The cooperativity of transition of U<–>I (U represents the ensemble of unfolded states) in apoMb at pH 4.0 is determined not only by the sequence but also by the anion concentration. Point mutations can substantially alter the cooperativity of formation of I. Another class of intermediates arise owing to bottlenecks in the rugged energy landscape that arises from topological frustration. As a result of the rough energy landscape, folding is predicted to follow the kinetic partitioning mechanism (KPM). According to KPM a fraction of molecules reaches the native state rapidly, while the remaining fraction is kinetically trapped in intermediates. The folding of lysozyme at pH 5.5 follows KPM. Our perspective also shows that the fraction of fast folding trajectories can be altered by changing pH, for example. These observations are clearly illustrated in simple off-lattice models of proteins. The simulations show that equilibrium intermediates occur “on-pathway” and have substantial probability to be revisited after the native state is reached, while kinetic intermediates are almost never sampled after native state is reached. In addition, kinetic intermediates are higher in free energy than equilibrium intermediates. We also discuss the consequences of multiple routes and intermediates on the transition state ensemble (TSE) in folding. Whenever multiple routes to the native state dominate, Phi-values can be larger than unity or less than zero. There appears to be a relationship between the diversity of structures in the denatured state ensemble and the extent to which the TSE is plastic. Simulations of beta-hairpins are used to illustrate these ideas.

Mechanical unfolding of RNA structures, ranging from hairpins to ribozymes, using laser optical tweezer experiments have begun to reveal the features of the energy landscape that cannot be easily explored using conventional experiments. Upon application of constant force (f), RNA hairpins undergo cooperative transitions from folded to unfolded states whereas subdomains of ribozymes unravel one at a time. Here, we use a self-organized polymer model and Brownian dynamics simulations to probe mechanical unfolding at constant force and constant-loading rate of four RNA structures of varying complexity. For simple hairpins, such as P5GA, application of constant force or constant loading rate results in bistable cooperative transitions between folded and unfolded states without populating any intermediates. The transition state location (DeltaxFTS) changes dramatically as the loading rate is varied. At loading rates comparable to those used in laser optical tweezer experiments, the hairpin is plastic, with DeltaxFTS being midway between folded and unfolded states; whereas at high loading rates, DeltaxFTS moves close to the folded state, i.e., RNA is brittle. For the 29-nucleotide TAR RNA with the three-nucleotide bulge, unfolding occurs in a nearly two-state manner with an occasional pause in a high free energy metastable state. Forced unfolding of the 55 nucleotides of the Hepatitis IRES domain IIa, which has a distorted L-shaped structure, results in well-populated stable intermediates. The most stable force-stabilized intermediate represents straightening of the L-shaped structure. For these structures, the unfolding pathways can be predicted using the contact map of the native structures. Unfolding of a RNA motif with internal multiloop, namely, the 109-nucleotide prohead RNA that is part of the 29 DNA packaging motor, at constant value of rf occurs with three distinct rips that represent unraveling of the paired helices. The rips represent kinetic barriers to unfolding. Our work shows 1), the response of RNA to force is largely determined by the native structure; and 2), only by probing mechanical unfolding over a wide range of forces can the underlying energy landscape be fully explored.

RNA molecules are exquisitely sensitive to the properties of counterions. The folding equilibrium of the Tetrahymena ribozyme is measured by nondenaturing gel electrophoresis in the presence of divalent group IIA metal cations. The stability of the folded ribozyme increases with the charge density (zeta) of the cation. Similar scaling is found when the free energy of the RNA folded in small and large metal cations is measured by urea denaturation. Brownian dynamics simulations of a polyelectrolyte show that the experimental observations can be explained by nonspecific ion-RNA interactions in the absence of site-specific metal chelation. The experimental and simulation results establish that RNA stability is largely determined by a combination of counterion charge and the packing efficiency of condensed cations that depends on the excluded volume of the cations.

Escherichia coli chaperonin, GroEL, helps proteins fold under nonpermissive conditions. During the reaction cycle, GroEL undergoes allosteric transitions in response to binding of a substrate protein (SP), ATP, and the cochaperonin GroES. Using coarse-grained representations of the GroEL and GroES structures, we explore the link between allosteric transitions and the folding of a model SP, a de novo-designed four-helix bundle protein, with low spontaneous yield. The ensemble of GroEL-bound SP is less structured than the bulk misfolded structures. Upon binding, which kinetically occurs in two stages, the SP loses not only native tertiary contacts but also experiences a decrease in helical content. During multivalent binding and the subsequent ATP-driven transition of GroEL the SP undergoes force-induced stretching. Upon encapsulation, which occurs upon GroES binding, the SP finds itself in a “hydrophilic” cavity in which it can reach the folded conformation. Surprisingly, we find that the yield of the native state in the expanded GroEL cavity is relatively small even after it remains in it for twice the spontaneous folding time. Thus, in accord with the iterative annealing mechanism, multiple rounds of binding, partial unfolding, and release of the SP are required to enhance the yield of the folded SP.

In order to clarify the mechanism of denaturant-induced unfolding of proteins we have calculated the interactions between hydrophobic and ionic species in aqueous guanidinium chloride and urea solutions using molecular dynamics simulations. Hydrophobic association is not significantly changed in urea or guanidinium chloride solutions. The strength of interaction between ion pairs is greatly diminished by the guanidinium ion. Although the changes in electrostatic interactions in urea are small, examination of structures, using appropriate pair functions, of urea and water around the solutes show strong hydrogen bonding between urea’s carbonyl oxygen and the positively charged solute. Our results strongly suggest protein denaturation occurs by the direct interaction model according to which the most commonly used denaturants unfold proteins by altering electrostatic interactions either by solvating the charged residues or by engaging in hydrogen bonds with the protein backbone. To further validate the direct interaction model we show that, in urea and guanidinium chloride solutions, unfolding of an unusually stable helix (H1) from mouse PrPC (residues 144-153) occurs by hydrogen bonding of denaturants to charged side chains and backbone carbonyl groups.

Nonfibrillar soluble oligomers, which are intermediates in the transition from monomers to amyloid fibrils, may be the toxic species in Alzheimer’s disease. To monitor the early events that direct assembly of amyloidogenic peptides we probe the dynamics of formation of (Abeta(16-22))(n) by adding a monomer to a preformed (Abeta(16-22))(n-1) (n = 4-6) oligomer in which the peptides are arranged in an antiparallel beta-sheet conformation. All atom molecular dynamics simulations in water and multiple long trajectories, for a cumulative time of 6.9 mus, show that the oligomer grows by a two-stage dock-lock mechanism. The largest conformational change in the added disordered monomer occurs during the rapid ( approximately 50 ns) first dock stage in which the beta-strand content of the monomer increases substantially from a low initial value. In the second slow-lock phase, the monomer rearranges to form in register antiparallel structures. Surprisingly, the mobile structured oligomers undergo large conformational changes in order to accommodate the added monomer. The time needed to incorporate the monomer into the fluid-like oligomer grows even when n = 6, which suggests that the critical nucleus size must exceed six. Stable antiparallel structure formation exceeds hundreds of nanoseconds even though frequent interpeptide collisions occur at elevated monomer concentrations used in the simulations. The dock-lock mechanism should be a generic mechanism for growth of oligomers of amyloidogenic peptides.

Escherichia coli dihydrofolate reductase (DHFR) catalyzes the reduction of dihydrofolate to tetrahydrofolate. During the catalytic cycle, DHFR undergoes conformational transitions between the closed (CS) and occluded (OS) states that, respectively, describe whether the active site is closed or occluded by the Met20 loop. The CS–>OS and the reverse transition may be viewed as allosteric transitions. Using a sequence-based approach, we identify a network of residues that represents the allostery wiring diagram. Many of the residues in the allostery wiring diagram, which are dispersed throughout the adenosine-binding domain as well as the loop domain, are not conserved. Several of the residues in the network have been previously shown by NMR experiments, mutational studies, and molecular dynamics simulations to be linked to equilibration conformational fluctuations of DHFR. To further probe the nature of events that occur during conformational fluctuations, we use a self-organized polymer model to monitor the kinetics of the CS–>OS and the reverse transitions. During the CS–>OS transition, coordinated changes in a number of residues in the loop domain enable the Met20 loop to slide along the alpha-helix in the adenosine-binding domain. Sliding is triggered by pulling of the Met20 loop by the betaG-betaH loop and the pushing action of the betaG-betaH loop. The residues that facilitate the Met20 loop motion are part of the network of residues that transmit allosteric signals during the CS–>OS transition. Replacement of M16 and G121, whose C(alpha) atoms are about 4.3 A in the CS, by a disulfide cross-link impedes that CS–>OS transition. The order of events in the OS–>CS transition is not the reverse of the forward transition. The contact Glu18-Ser49 in the OS persists until the sliding of the Met20 loop is nearly complete. The ensemble of structures in the transition state in both the allosteric transitions is heterogeneous. The most probable transition-state structure resembles the OS (CS) in the CS–>OS (OS–>CS) transition, which is in accord with the Hammond postulate. Structures resembling the OS (CS) are present as minor ( approximately 1-3%) components in equilibrated CS (OS) structures.

The Escherichia coli chaperonin GroEL, which helps proteins to fold, consists of two heptameric rings stacked back-to-back. During the reaction cycle GroEL undergoes a series of allosteric transitions triggered by ligand (substrate protein, ATP, and the cochaperonin GroES) binding. Based on an elastic network model of the bullet-shaped double-ring chaperonin GroEL-(ADP)(7)-GroES structure (R”T state), we perform a normal mode analysis to explore the energetically favorable collective motions encoded in the R”T structure. By comparing each normal mode with the observed conformational changes in the R”T –> TR” transition, a single dominant normal mode provides a simple description of this highly intricate allosteric transition. A detailed analysis of this relatively high-frequency mode describes the structural and dynamic changes that underlie the positive intra-ring and negative inter-ring cooperativity. The dynamics embedded in the dominant mode entails highly concerted structural motions with approximate preservation of sevenfold symmetry within each ring and negatively correlated ones between the two rings. The dominant normal mode (in comparison with the other modes) is robust to parametric perturbations caused by sequence variations, which validates its functional importance. Response of the dominant mode to local changes that mimic mutations using the structural perturbation method technique leads to a wiring diagram that identifies a network of key residues that regulate the allosteric transitions. Many of these residues are located in intersubunit interfaces, and may therefore play a critical role in transmitting allosteric signals between subunits.

Characterization of the structures of the transition state ensemble is a key step in describing the folding reaction. Using two variants of a coarse-grained model of the three-stranded beta-sheet WW domain and a fully automated progress variable clustering (PVC) algorithm, we have dissected the effect of macromolecular crowding and confinement on the changes in the transition state structures in comparison to bulk. Each amino acid is represented using a Calpha atom and a side chain. The distance between the Calpha atom and center of mass of the side chain is taken to be its effective van der Waals radius. For the bulk case, we predict using the PVC algorithm, which does not assume knowledge of the underlying folding reaction coordinate, that there are two classes of structures in the transition state ensemble (TSE). The structures in both of the classes are compact. The dominant cluster is more structured than the structures in the less populated class. In accord with bulk experiments, the residues in strands beta2 and beta3 and the interactions at the beta2-beta3 interface are structured. When only excluded volume interactions between the crowding particles and the WW domain are taken into account or when the protein is confined to an inert spherical pore, the overall structure of the TSE does not change dramatically. However, in this entropy dominated regime, the width of the TSE decreases and the structures become more oblate and less spherical as the volume fraction of crowding particle increases or when the pore radius decreases. It suggests that the shape changes, which are computed using the moment of inertia tensor, may represent the slow degrees of freedom during the folding process. When non-native interactions between side chains and interactions with the cavity of the pores are taken into account, the TSE becomes considerably broader. Although the topology in the transition has a fold similar to the native state, the structures are far more plastic than in the bulk. The TSE is sensitive to the size of the pore as well as interactions between the pore and the protein. The differences between the two cases (confinement in an inert pore and when pore-protein interactions are considered) arise due to the increased importance of enthalpic interactions in the transition state as the strength of the protein-pore interaction increases.

Nanomanipulation of biomolecules by using single-molecule methods and computer simulations has made it possible to visualize the energy landscape of biomolecules and the structures that are sampled during the folding process. We use simulations and single-molecule force spectroscopy to map the complex energy landscape of GFP that is used as a marker in cell biology and biotechnology. By engineering internal disulfide bonds at selected positions in the GFP structure, mechanical unfolding routes are precisely controlled, thus allowing us to infer features of the energy landscape of the wild-type GFP. To elucidate the structures of the unfolding pathways and reveal the multiple unfolding routes, the experimental results are complemented with simulations of a self-organized polymer (SOP) model of GFP. The SOP representation of proteins, which is a coarse-grained description of biomolecules, allows us to perform forced-induced simulations at loading rates and time scales that closely match those used in atomic force microscopy experiments. By using the combined approach, we show that forced unfolding of GFP involves a bifurcation in the pathways to the stretched state. After detachment of an N-terminal alpha-helix, unfolding proceeds along two distinct pathways. In the dominant pathway, unfolding starts from the detachment of the primary N-terminal beta-strand, while in the minor pathway rupture of the last, C-terminal beta-strand initiates the unfolding process. The combined approach has allowed us to map the features of the complex energy landscape of GFP including a characterization of the structures, albeit at a coarse-grained level, of the three metastable intermediates.

Allosteric interactions between residues that are spatially apart and well separated in sequence are important in the function of multimeric proteins as well as single-domain proteins. This observation suggests that, among the residues that are involved in long-range communications, mutation at one site should affect interactions at a distant site. By adopting a sequence-based approach, we present an automated approach that uses a generalization of the familiar sequence entropy in conjunction with a coupled two-way clustering algorithm, to predict the network of interactions that trigger allosteric interactions in proteins. We use the method to identify the subset of dynamically important residues in three families, namely, the small PDZ family, G protein-coupled receptors (GPCR), and the Lectins, which are cell-adhesion receptors that mediate the tethering and rolling of leukocytes on inflamed endothelium. For the PDZ and GPCR families, our procedure predicts, in agreement with previous studies, a network containing a small number of residues that are involved in their function. Application to the Lectin family reveals a network of residues interspersed throughout the C-terminal end of the structure that are responsible for binding to ligands. Based on our results and previous studies, we propose that functional robustness requires that only a small subset of distantly connected residues be involved in transmitting allosteric signals in proteins.

Dynamics of tRNA was studied using neutron scattering spectroscopy. Despite vast differences in the architecture and backbone structure of proteins and RNA, hydrated tRNA undergoes the dynamic transition at the same temperature as hydrated lysozyme. The similarity of the dynamic transition in RNA and proteins supports the idea that it is solvent induced. Because tRNA essentially has no methyl groups, the results also suggest that methyl groups are not the main contributor of the dynamic transition in biological macromolecules. However, they may explain strong differences in the dynamics of tRNA and lysozyme observed at low temperatures.

We analyze the dependence of cooperativity of the thermal denaturation transition and folding rates of globular proteins on the number of amino acid residues, N, using lattice models with side chains, off-lattice Go models, and the available experimental data. A dimensionless measure of cooperativity, Omega(c) (0 < Omega(c) < infinity), scales as Omega(c) approximately N(zeta). The results of simulations and the analysis of experimental data further confirm the earlier prediction that zeta is universal with zeta = 1 + gamma, where exponent gamma characterizes the susceptibility of a self-avoiding walk. This finding suggests that the structural characteristics in the denaturated state are manifested in the folding cooperativity at the transition temperature. The folding rates k(F) for the Go models and a dataset of 69 proteins can be fit using k(F) = k(F)0 exp(-cN(beta)). Both beta = 1/2 and 2/3 provide a good fit of the data. We find that k(F) = k(F)0 exp(-cN(1/2)), with the average (over the dataset of proteins) k(F)0 approximately (0.2 micros)(-1) and c approximately 1.1, can be used to estimate folding rates to within an order of magnitude in most cases. The minimal models give identical N dependence with c approximately 1. The prefactor for off-lattice Go models is nearly 4 orders of magnitude larger than the experimental value.

Nanomanipulation of individual RNA molecules, using laser optical tweezers, has made it possible to infer the major features of their energy landscape. Time-dependent mechanical unfolding trajectories, measured at a constant stretching force (f(S)) of simple RNA structures (hairpins and three-helix junctions) sandwiched between RNA/DNA hybrid handles show that they unfold in a reversible all-or-none manner. To provide a molecular interpretation of the experiments we use a general coarse-grained off-lattice Gō-like model, in which each nucleotide is represented using three interaction sites. Using the coarse-grained model we have explored forced-unfolding of RNA hairpin as a function of f(S) and the loading rate (r(f)). The simulations and theoretical analysis have been done both with and without the handles that are explicitly modeled by semiflexible polymer chains. The mechanisms and timescales for denaturation by temperature jump and mechanical unfolding are vastly different. The directed perturbation of the native state by f(S) results in a sequential unfolding of the hairpin starting from their ends, whereas thermal denaturation occurs stochastically. From the dependence of the unfolding rates on r(f) and f(S) we show that the position of the unfolding transition state is not a constant but moves dramatically as either r(f) or f(S) is changed. The transition-state movements are interpreted by adopting the Hammond postulate for forced-unfolding. Forced-unfolding simulations of RNA, with handles attached to the two ends, show that the value of the unfolding force increases (especially at high pulling speeds) as the length of the handles increases. The pathways for refolding of RNA from stretched initial conformation, upon quenching f(S) to the quench force f(Q), are highly heterogeneous. The refolding times, upon force-quench, are at least an order-of-magnitude greater than those obtained by temperature-quench. The long f(Q)-dependent refolding times starting from fully stretched states are analyzed using a model that accounts for the microscopic steps in the rate-limiting step, which involves the trans to gauche transitions of the dihedral angles in the GAAA tetraloop. The simulations with explicit molecular model for the handles show that the dynamics of force-quench refolding is strongly dependent on the interplay of their contour length and persistence length and the RNA persistence length. Using the generality of our results, we also make a number of precise experimentally testable predictions.

Mechanical folding trajectories for polyproteins starting from initially stretched conformations generated by single-molecule atomic force microscopy experiments [Fernandez, J. M. & Li, H. (2004) Science 303, 1674-1678] show that refolding, monitored by the end-to-end distance, occurs in distinct multiple stages. To clarify the molecular nature of folding starting from stretched conformations, we have probed the folding dynamics, upon force quench, for the single I27 domain from the muscle protein titin by using a C(alpha)-Go model. Upon temperature quench, collapse and folding of I27 are synchronous. In contrast, refolding from stretched initial structures not only increases the folding and collapse time scales but also decouples the two kinetic processes. The increase in the folding times is associated primarily with the stretched state to compact random coil transition. Surprisingly, force quench does not alter the nature of the refolding kinetics, but merely increases the height of the free-energy folding barrier. Force quench refolding times scale as tau(F) approximately tau(F)(0)exp(f(q)Deltax(f)/k(B)T), where Deltax(f) approximately 0.6 nm is the location of the average transition state along the reaction coordinate given by end-to-end distance. We predict that tau(F) and the folding mechanism can be dramatically altered by the initial and/or final values of force. The implications of our results for design and analysis of experiments are discussed.

We have studied the stability and the yield of the folded WW domains in a spherical nanopore to provide insights into the changes in the folding characteristics due to interactions of the polypeptide (SP) with the walls of the pore. Using different models for the interactions between the nanopore and the polypeptide chain we have obtained results that are relevant to a broad range of experiments. (a) In the temperature and the strength of the SP-pore interaction plane (lambda), there are four “phases,” namely, the unfolded state, the native state, the molten globule phase (MG), and the surface interaction-stabilized (SIS) state. The MG and SIS states are populated at moderate and large values of lambda, respectively. For a fixed pore size, the folding rates vary non-monotonically as lambda is varied with a maximum at lambda approximately 1 at which the SP-nanopore interaction is comparable to the stability of the native state. At large lambda values, the WW domain is kinetically trapped in the SIS states. Using multiple sequence alignment, we conclude that similar folding mechanism should be observed in other WW domains as well. (b) To mimic the changes in the nature of the allosterically driven SP-GroEL interactions we consider two models for the dynamic Anfinsen cage (DAC). In DAC1, the SP-cavity interaction cycles between hydrophobic (lambda>0) and hydrophilic (lambda=0) with a period tau. The yield of the native state is a maximum for an optimum value of tau=tau(OPT). At tau=tau(OPT), the largest yield of the native state is obtained when tau(H) approximately tau(P) where tau(H)(tau(P)) is the duration for which the cavity is hydrophobic (hydrophilic). Thus, in order to enhance the native state yield, the cycling rate, for a given loading rate of the GroEL nanomachine, should be maximized. In DAC2, the volume of the cavity is doubled (as happens when ATP and GroES bind to GroEL) and the SP-pore interaction simultaneously changes from hydrophobic to hydrophilic. In this case, we find greater increase in yield of the native state compared to DAC1 at all values of tau.

The self-assembly of RNA structure depends on the interactions of counterions with the RNA and with each other. Comparison of various polyamines showed that the tertiary structure of the Tetrahymena ribozyme is more stable when the counterions are small and highly charged. By monitoring the folding kinetics of the ribozyme as a function of polyamine concentration, we now find that the charge density of the counterions determines the positions of the folding transition states. The transition state ensemble (TSE) between U and N moves away from the native state as the counterion valence and charge density increase, as predicted by the Hammond postulate. The TSE is broader and less structured when the RNA is refolded in polyamines rather than Mg2+. That the charge density of the counterions determines the plasticity of the TSE demonstrates the importance of interactions among condensed counterions for the self-assembly of RNA structures. We propose that the major barrier to RNA folding is dominated by entropy changes when counterion charge density is low and enthalpy differences when it is high.

A number of situations such as protein folding in confined spaces, lubrication in tight spaces, and chemical reactions in confined spaces require an understanding of water-mediated interactions. As an illustration of the profound effects of confinement on hydrophobic and ionic interactions, we investigate the solvation of methane and methane decorated with charges in spherically confined water droplets. Free energy profiles for a single methane molecule in droplets, ranging in diameter (D) from 1 to 4 nm, show that the droplet surfaces are strongly favorable as compared to the interior. From the temperature dependence of the free energy in D = 3 nm, we show that this effect is entropically driven. The potentials of mean force (PMFs) between two methane molecules show that the solvent separated minimum in the bulk is completely absent in confined water, independent of the droplet size since the solute particles are primarily associated with the droplet surface. The tendency of methanes with charges (M(q+) and M(q-) with q(+) = |q(-)| = 0.4e, where e is the electronic charge) to be pinned at the surface depends dramatically on the size of the water droplet. When D = 4 nm, the ions prefer the interior whereas for D < 4 nm the ions are localized at the surface, but with much less tendency than for methanes. Increasing the ion charge to e makes the surface strongly unfavorable. Reflecting the charge asymmetry of the water molecule, negative ions have a stronger preference for the surface compared to positive ions of the same charge magnitude. With increasing droplet size, the PMFs between M(q+) and M(q-) show decreasing influence of the boundary owing to the reduced tendency for surface solvation. We also show that as the solute charge density decreases the surface becomes less unfavorable. The implications of our results for the folding of proteins in confined spaces are outlined.

Loop formation between monomers in the interior of semiflexible chains describes elementary events in biomolecular folding and DNA bending. We calculate analytically the interior distance distribution function for semiflexible chains using a mean field approach. Using the potential of mean force derived from the distance distribution function we present a simple expression for the kinetics of interior looping by adopting Kramers theory. For the parameters, that are appropriate for DNA, the theoretical predictions in comparison with the case are in excellent agreement with explicit Brownian dynamics simulations of wormlike chain (WLC) model. The interior looping times (tauIC) can be greatly altered in the cases when the stiffness of the loop differs from that of the dangling ends. If the dangling end is stiffer than the loop then tauIC increases for the case of the WLC with uniform persistence length. In contrast, attachment of flexible dangling ends enhances rate of interior loop formation. The theory also shows that if the monomers are charged and interact via screened Coulomb potential then both the cyclization (tauc) and interior looping (tauIC) times greatly increase at low ionic concentration. Because both tauc and tauIC are determined essentially by the effective persistence length [lp(R)] we computed lp(R) by varying the range of the repulsive interaction between the monomers. For short range interactions lp(R) nearly coincides with the bare persistence length which is determined largely by the backbone chain connectivity. This finding rationalizes the efficacy of describing a number of experimental observations (response of biopolymers to force and cyclization kinetics) in biomolecules using WLC model with an effective persistence length.

By representing the high-resolution crystal structures of a number of enzymes using the elastic network model, it has been shown that only a few low-frequency normal modes are needed to describe the large-scale domain movements that are triggered by ligand binding. Here we explore a link between the nearly invariant nature of the modes that describe functional dynamics at the mesoscopic level and the large evolutionary sequence variations at the residue level. By using a structural perturbation method (SPM), which probes the residue-specific response to perturbations (or mutations), we identify a sparse network of strongly conserved residues that transmit allosteric signals in three structurally unrelated biological nanomachines, namely, DNA polymerase, myosin motor, and the Escherichia coli chaperonin. Based on the response of every mode to perturbations, which are generated by interchanging specific sequence pairs in a multiple sequence alignment, we show that the functionally relevant low-frequency modes are most robust to sequence variations. Our work shows that robustness of dynamical modes at the mesoscopic level is encoded in the structure through a sparse network of residues that transmit allosteric signals.

We present, to our knowledge, a new theory that takes internal dynamics of proteins into account to describe forced-unfolding and force-quench refolding in single molecule experiments. In the current experimental setup (using either atomic force microscopy or laser optical tweezers) the distribution of unfolding times, P(t), is measured by applying a constant stretching force f(S) from which the apparent f(S)-dependent unfolding rate is obtained. To describe the complexity of the underlying energy landscape requires additional probes that can incorporate the dynamics of tension propagation and relaxation of the polypeptide chain upon force quench. We introduce a theory of force correlation spectroscopy to map the parameters of the energy landscape of proteins. In force correlation spectroscopy, the joint distribution P(T, t) of folding and unfolding times is constructed by repeated application of cycles of stretching at constant f(S) separated by release periods T during which the force is quenched to f(Q) < f(S). During the release period, the protein can collapse to a manifold of compact states or refold. We show that P(T, t) at various f(S) and f(Q) values can be used to resolve the kinetics of unfolding as well as formation of native contacts. We also present methods to extract the parameters of the energy landscape using chain extension as the reaction coordinate and P(T, t). The theory and a wormlike chain model for the unfolded states allows us to obtain the persistence length l(p) and the f(Q)-dependent relaxation time, giving us an estimate of collapse timescale at the single molecular level, in the coil states of the polypeptide chain. Thus, a more complete description of landscape of protein native interactions can be mapped out if unfolding time data are collected at several values of f(S) and f(Q). We illustrate the utility of the proposed formalism by analyzing simulations of unfolding-refolding trajectories of a coarse-grained protein (S1) with beta-sheet architecture for several values of f(S), T, and f(Q) = 0. The simulations of stretch-relax trajectories are used to map many of the parameters that characterize the energy landscape of S1.

We have used a bioinformatic approach to predict the natural substrate proteins for the Escherichia coli chaperonin GroEL based on two simple criteria. Natural substrate proteins should contain binding motifs similar in sequence to the mobile loop peptide of GroES that displaces the binding motif during the chaperonin cycle. Secondly, each substrate protein should contain multiple copies of the binding motif so that the chaperonin can perform “work” on the substrate protein. To validate these criteria, we have used a database of 252 proteins that have been experimentally shown to interact with the chaperonin machinery in vivo. More than 80% are identified by these criteria. The binding motifs of all 79 proteins in the database with a known three-dimensional structure are buried (<50% solvent-accessible surface area) in the native state. Our results show that the binding motifs are inaccessible in the native state but become solvent-exposed in unfolded state, thus enabling GroEL to distinguish between unfolded and native states. The structures of the binding motif in the native states of the substrate proteins include alpha-helices, beta-strands, and random coils. The diversity of secondary structures implies that there are large and varied conformational transitions in the recognition motifs after their displacement by the mobile loops of GroES.

Systematic studies based on 1H NMR and 13C NMR indicated that the alkylthio group behaves as a weak electron-withdrawing group in a simple aniline system like 2-butylthioaniline, while the same alkylthio group clearly acted as a resonance electron-donating group in higher conjugated aniline trimer systems, like butylthio-substituted PDA (mono-PDA) and dibutylthio-substituted PDA (2,6-diPDA). The formation of 2,6-diPDA as the major byproduct during the preparation of mono-PDA from PDI and butane-1-thiol provided additional support for the resonance electron donating nature of the butylthio group in these aniline trimer systems. Furthermore, CV studies also clearly indicated that the redox potential E degrees (vs. SCE) of the aniline trimer systems decreased with the increase in the number of butylthio groups, further confirming the electron-donating nature of the butylthio group in these higher conjugated trimer systems.

Atomic force measurements of unbinding rates (or off-rates) of ligands bound to a class of cell adhesion molecules from the selectin family show a transition from catch to slip bonds as the value of external force (f) is increased. At low forces (<10 pN), the unbinding rates decrease (catch regime), while, at high forces, the rates increase in accord with the Bell model (slip regime). The energy landscape underlying the catch-slip transition can be captured by a two-state model that considers the possibility of redistribution of population from the force-free bound state to the force-stabilized bound state. The excellent agreement between theory and experiments is used to extract the parameters characterizing the energy landscape of the complex by fitting the calculated curves to lifetime data (obtained at constant f) for the monomeric form of PSGL-1 (sPSGL-1). We used the constant force parameters to predict the distributions of unbinding times and unbinding forces as a function of the loading rate. The general two-state model, which also correctly predicts the absence of catch bonds in the binding of antibodies to selectins, is used to resolve the energy landscape parameters characterizing adhesive interactions of P- and L-selectins with physiological ligands such as sPSGL-1 and endoglycan and antibodies such as G1 and DREG56. Despite high sequence similarity, the underlying shapes of the energy landscape of P-selectin and L-selectin interacting with sPSGL-1 are markedly different. The underlying energy landscape of the selectin cell adhesion complex is sensitive to the nature of the ligand. The unified description of selectins bound to physiological ligands and antibodies in conjunction with experimental data can be used to extract the key parameters that describe the dynamics of cell adhesion complexes.

The chaperonin GroEL-GroES, a machine that helps proteins to fold, cycles through a number of allosteric states, the T state, with high affinity for substrate proteins, the ATP-bound R state, and the R” (GroEL-ADP-GroES) complex. Here, we use a self-organized polymer model for the GroEL allosteric states and a general structure-based technique to simulate the dynamics of allosteric transitions in two subunits of GroEL and the heptamer. The T –> R transition, in which the apical domains undergo counterclockwise motion, is mediated by a multiple salt-bridge switch mechanism, in which a series of salt-bridges break and form. The initial event in the R –>R” transition, during which GroEL rotates clockwise, involves a spectacular outside-in movement of helices K and L that results in K80-D359 salt-bridge formation. In both the transitions there is considerable heterogeneity in the transition pathways. The transition state ensembles (TSEs) connecting the T, R, and R” states are broad with the TSE for the T –> R transition being more plastic than the R –> R” TSE.

In the amyloid fibrils formed from long fragments of the amyloid beta-protein (Abeta-protein), the monomers are arranged in parallel and lie perpendicular to the fibril axis. The structure of the monomers satisfies the amyloid self-organization principle; namely, the low free energy state of the monomer maximizes the number of intra- and interpeptide contacts and salt bridges. The formation of the intramolecular salt bridge between Asp(D)23 and Lys(K)28 ensures that unpaired charges are not buried in the low-dielectric interior. We have investigated, using all-atom molecular dynamics simulations in explicit water, whether the D23-K28 interaction forms spontaneously in the isolated Abeta10-35 monomer. To validate the simulation protocol, we show, using five independent trajectories spanning a total of 100 ns, that the pKa values of the titratable groups are in good agreement with experimental measurements. The computed free energy disconnectvity graph shows that broadly the ensemble of compact random coil conformations can be clustered into four basins that are separated by free energy barriers ranging from 0.3 to 2.7 kcal/mol. There is significant residual structure in the conformation of the peptide in each of the basins. Due to the desolvation penalty, the structural motif with a stable turn involving the residues VGSN and a preformed D23-K28 contact is a minor component of the simulated structures. The extent of solvation of the peptides in the four basins varies greatly, which underscores the dynamical fluctuations in the monomer. Our results suggest that the early event in the oligomerization process must be the expulsion of discrete water molecules that facilitates the formation of interpeptide-interaction-driven stable structures with an intramolecular D23-K28 salt bridge and an intact VGSN turn.

Using self-organized polymer models, we predict mechanical unfolding and refolding pathways of ribozymes, and the green fluorescent protein. In agreement with experiments, there are between six and eight unfolding transitions in the Tetrahymena ribozyme. Depending on the loading rate, the number of rips in the force-ramp unfolding of the Azoarcus ribozymes is between two and four. Force-quench refolding of the P4-P6 subdomain of the Tetrahymena ribozyme occurs through a compact intermediate. Subsequent formation of tertiary contacts between helices P5b-P6a and P5a/P5c-P4 leads to the native state. The force-quench refolding pathways agree with ensemble experiments. In the dominant unfolding route, the N-terminal alpha helix of GFP unravels first, followed by disruption of the N terminus beta strand. There is a third intermediate that involves disruption of three other strands. In accord with experiments, the force-quench refolding pathway of GFP is hierarchic, with the rate-limiting step being the closure of the barrel.

Determination of sizes and flexibilities of RNA molecules is important in understanding the nature of packing in folded structures and in elucidating interactions between RNA and DNA or proteins. Using the coordinates of the structures of RNA in the Protein Data Bank we find that the size of the folded RNA structures, measured using the radius of gyration R(G), follows the Flory scaling law, namely, R(G)=5.5N(1/3) A, where N is the number of nucleotides. The shape of RNA molecules is characterized by the asphericity Delta and the shape S parameters that are computed using the eigenvalues of the moment of inertia tensor. From the distribution of Delta, we find that a large fraction of folded RNA structures are aspherical and the distribution of S values shows that RNA molecules are prolate (S>0). The flexibility of folded structures is characterized by the persistence length l(p). By fitting the distance distribution function P(r), that is computed using the coordinates of the folded RNA, to the wormlike chain model we extracted the persistence length l(p). We find that l(p) approximately 1.5N(0.33) A which might reflect the large separation between the free energies that stabilize secondary and tertiary structures. The dependence of l(p) on N implies that the average length of helices should increase as the size of RNA grows. We also analyze packing in the structures of ribosomes (30S, 50S, and 70S) in terms of R(G), Delta, S, and l(p). The 70S and the 50S subunits are more spherical compared to most RNA molecules. The globularity in 50S is due to the presence of an unusually large number (compared to 30S subunit) of small helices that are stitched together by bulges and loops. Comparison of the shapes of the intact 70S ribosome and the constituent particles suggests that folding of the individual molecules might occur prior to assembly.

The Escherichia coli chaperonin machinery, GroEL, assists the folding of a number of proteins. We describe a sequence-based approach to identify the natural substrate proteins (SPs) for GroEL. Our method is based on the hypothesis that natural SPs are those that contain patterns of residues similar to those found in either GroES mobile loop and/or strongly binding peptide in complex with GroEL. The method is validated by comparing the predicted results with experimentally determined natural SPs for GroEL. We have searched for such patterns in five genomes. In the E. coli genome, we identify 1422 (about one-third) sequences that are putative natural SPs. In Saccharomyces cerevisiae, 2885 (32%) of sequences can be natural substrates for Hsp60, which is the analog of GroEL. The precise number of natural SPs is shown to be a function of the number of contacts an SP makes with the apical domain (N(C)) and the number of binding sites (N(B)) in the oligomer with which it interacts. For known SPs for GroEL, we find approximately 4 < N(C) < 5 and 2

Characterization of the early stages of peptide aggregation is of fundamental importance in elucidating the mechanism of the formation of deposits associated with amyloid disease. The initial step in the pathway of aggregation of the Abeta-protein, whose monomeric NMR structure is known, was studied through the simulation of the structure and stability of the peptide dimer in aqueous solution. A protocol based on shape complementarity was used to generate an assortment of possible dimer structures. The structures generated based on shape complementarity were evaluated using rapidly computed estimates of the desolvation and electrostatic interaction energies to identify a putative stable dimer structure. The potential of mean force associated with the dimerization of the peptides in aqueous solution was computed for both the hydrophobic and the electrostatic driven forces using umbrella sampling and classical molecular dynamics simulation at constant temperature and pressure with explicit solvent and periodic boundary conditions. The comparison of the two free energy profiles suggests that the structure of the peptide dimer is determined by the favorable desolvation of the hydrophobic residues at the interface. Molecular dynamics trajectories originating from two putative dimer structures indicate that the peptide dimer is stabilized primarily through hydrophobic interactions, while the conformations of the peptide monomers undergo substantial structural reorganization in the dimerization process. The finding that the phi-dimer may constitute the ensemble of stable Abeta(10-35) dimer has important implications for fibril formation. In particular, the expulsion of water molecules at the interface might be a key event, just as in the oligomerization of Abeta(16-22) fragments. We conjecture that events prior to the nucleation process themselves might involve crossing free energy barriers which depend on the peptide-peptide and peptide-water interactions. Consistent with existing experimental studies, the peptides within the ensemble of aggregated states show no signs of formation of secondary structure.

The unbinding dynamics of complexes involving cell-adhesion molecules depends on the specific ligands. Atomic force microscopy measurements have shown that for the specific P-selectin-P-selectin glycoprotein ligand (sPSGL-1) the average bond lifetime t initially increases (catch bonds) at low (< or =10 pN) constant force, f, and decreases when f > 10 pN (slip bonds). In contrast, for the complex with G1 anti-P-selectin monoclonal antibody t monotonically decreases with f. To quantitatively map the energy landscape of such complexes we use a model that considers the possibility of redistribution of population from one force-free state to another force-stabilized bound state. The excellent agreement between theory and experiments allows us to extract energy landscape parameters by fitting the calculated curves to the lifetime measurements for both sPSGL-1 and G1. Surprisingly, the unbinding transition state for P-selectin-G1 complex is close (0.32 nm) to the bound state, implying that the interaction is brittle, i.e., once deformed, the complex fractures. In contrast, the unbinding transition state of the P-selectin-sPSGL-1 complex is far (approximately 1.5 nm) from the bound state, indicative of a compliant structure. Constant f energy landscape parameters are used to compute the distributions of unbinding times and unbinding forces as a function of the loading rate, rf. For a given rf, unbinding of sPSGL-1 occurs over a broader range of f with the most probable f being an order of magnitude less than for G1. The theory for cell adhesion complexes can be used to predict the outcomes of unbinding of other protein-protein complexes.

A crucial step in the determination of the three-dimensional native structures of RNA is the prediction of their secondary structures, which are stable independent of the tertiary fold. Accurate prediction of the secondary structure requires context-dependent estimates of the interaction parameters. We have exploited the growing database of natively folded RNA structures in the Protein Data Bank (PDB) to obtain stacking interaction parameters using a knowledge-based approach. Remarkably, the calculated values of the resulting statistical potentials (SPs) are in excellent agreement with the parameters determined using measurements in small oligonucleotides. We validate the SPs by predicting 74% of the base-pairs in a dataset of structures using the ViennaRNA package. Interestingly, this number is similar to that obtained using the measured thermodynamic parameters. We also tested the efficacy of the SP in predicting secondary structure by using gapless threading, which we advocate as an alternative method for rapidly predicting RNA structures. For RNA molecules with less than 700 nucleotides, about 70% of the native base-pairs are correctly predicted. As a further validation of the SPs we calculated Z-scores, which measure the relative stability of the native state with respect to a manifold of higher free energy states. The computed Z-scores agree with estimates made using calorimetric measurements for a few RNA molecules. Structural analysis was used to rationalize the success and failures of SP and experimentally determined parameters. First, from the near perfect linear relationship between the number of native base-pairs and sequence length, we show that nearly 46% of nucleotides are not in stacks. Second, by analyzing the suboptimal structures that are generated in gapless threading we show that the SPs and experimentally determined parameters are most successful in predicting stacks that end in hairpins. These results show that further improvement in secondary structure prediction requires reliable estimates of interaction parameters for loops, bulges, and stacks that do not end in hairpins.

Mechanical unfolding trajectories, generated by applying constant force in optical-tweezer experiments, show that RNA hairpins and the P5abc subdomain of the group I intron unfold reversibly. We use coarse-grained Go-like models for RNA hairpins to explore forced unfolding over a broad range of temperatures. A number of predictions that are amenable to experimental tests are made. At the critical force, the hairpin jumps between folded and unfolded conformations without populating any discernible intermediates. The phase diagram in the force-temperature (f, T) plane shows that the hairpin unfolds by an all-or-none process. The cooperativity of the unfolding transition increases dramatically at low temperatures. Free energy of stability, obtained from time averages of mechanical unfolding trajectories, coincides with ensemble averages, which establishes ergodicity. The hopping time between the native basin of attraction (NBA) and the unfolded basin increases dramatically along the phase boundary. Thermal unfolding is stochastic, whereas mechanical unfolding occurs in “quantized steps” with great variations in the step lengths. Refolding times, upon force quench, from stretched states to the NBA are at least an order of magnitude greater than folding times by temperature quench. Upon force quench from stretched states, the NBA is reached in at least three stages. In the initial stages, the mean end-to-end distance decreases nearly continuously, and there is a sudden transition to the NBA only in the last stage. Because of the generality of the results, we propose that similar behavior should be observed in force quench refolding of proteins.

The presence of macromolecules in cells geometrically restricts the available space for poplypeptide chains. To study the effects of macromolecular crowding on folding thermodynamics and kinetics, we used an off-lattice model of the all-beta-sheet WW domain in the presence of large spherical particles whose interaction with the polypeptide chain is purely repulsive. At all volume fractions, phi(c), of the crowding agents the stability of the native state is enhanced. Remarkably, the refolding rates, which are larger than the value at phi(c) = 0, increase nonmonotonically as phi(c) increases, reaching a maximum at phi(c)=phi(c)(*). At high values of phi(c), the depletion-induced intramolecular attraction produces compact structures with considerable structure in the denatured state. Changes in native state stability and folding kinetics at phi(c) can be quantitatively mapped onto confinement in a volume-fraction-dependent spherical pore with radius R(s) approximately (4pi/3phi(c))(1/3) R(c) (R(c) is the radius of the crowding particles) as long as phi(c)< or =phi(c)(*). We show that the extent of native state stabilization at finite phi(c) is comparable with that in a spherical pore. In both situations, rate enhancement is due to destabilization of the denatured states with respect to phi(c) = 0.

The open/closed transition in polymerases is a crucial event in DNA replication and transcription. We hypothesize that the residues that transmit the signal for the open/closed transition are also strongly conserved. To identify the dynamically relevant residues, we use an elastic network model of polymerases and probe the residue-specific response to a local perturbation. In a variety of DNA/RNA polymerases, a network of residues spanning the fingers and palm domains is involved in the open/closed transition. The similarity in the network of residues responsible for large-scale domain movements supports the notion of a common induced-fit mechanism in the polymerase families for the formation of a closed ternary complex. Multiple sequence alignment shows that many of these residues are also strongly conserved. Residues with the largest sensitivity to local perturbations include those that are not so obviously involved in the polymerase catalysis. Our results suggest that mutations of the mechanical “hot spots” can compromise the efficiency of the enzyme.

A novel colorimetric and fluorescent chemosensor ADDTU-1 bearing dual receptor sites, which shows specific optical signaling for AcO-, H2PO4-, and F- over other anions and dual response toward AcO- and F- via PET and ICT mechanisms, is described. [structure: see text]

Although the intact chaperonin machinery is needed to rescue natural substrate proteins (SPs) under non-permissive conditions the “minichaperone” alone, containing only the isolated apical domain of GroEL, can assist folding of a certain class of proteins. To understand the annealing function of the minichaperone, we have carried out molecular dynamics simulations in the NPT ensemble totaling 300ns for four systems; namely, the isolated strongly binding peptide (SBP), the minichaperone, and the SBP and a weakly binding peptide (WBP) in complex with the minichaperone. The SBP, which is structureless in isolation, adopts a beta-hairpin conformation in complex with the minichaperone suggesting that favorable non-specific interactions of the SPs confined to helices H and I of the apical domains can induce local secondary structures. Comparison of the dynamical fluctuations of the apo and the liganded forms of the minichaperone shows that the stability (needed for SP capture) involves favorable hydrophobic interactions and hydrogen bond network formation between the SBP and WBP, and helices H and I. The release of the SP, which is required for the annealing action, involves water-mediated interactions of the charged residues at the ends of H and I helices. The simulation results are consistent with a transient binding release (TBR) model for the annealing action of the minichaperone. According to the TBR model, SP annealing occurs in two stages. In the first stage the SP is captured by the apical domain. This is followed by SP release (by thermal fluctuations) that places it in a different region of the energy landscape from which it can partition rapidly to the native state with probability Phi or be trapped in another misfolded state. The process of binding and release can result in enhancement of the native state yield. The TBR model suggests “that any cofactor that can repeatedly bind and release SPs can be effective in assisting protein folding.” By comparing the structures of the non-chaperone alpha-casein (which has no sequence similarity with the apical domain) and the minichaperone and the hydrophobicity profiles we show that alpha-casein has a pair of helices that have similar sequence and structural profiles as H and I. Based on this comparison we identify residues that stabilize (destabilize) alpha-casein-protein complexes. This suggests that alpha-casein assists folding by the TBR mechanism.

Visualizing the navigation of an ensemble of unfolded molecules through the bumpy energy landscape in search of the native state gives a pictorial view of biomolecular folding. This picture, when combined with concepts in polymer theory, provides a unified theory of RNA and protein folding. Just as for proteins, the major folding free energy barrier for RNA scales sublinearly with the number of nucleotides, which allows us to extract the elusive prefactor for RNA folding. Several folding scenarios can be anticipated by considering variations in the energy landscape that depend on sequence, native topology, and external conditions. RNA and protein folding mechanism can be described by the kinetic partitioning mechanism (KPM) according to which a fraction (Phi) of molecules reaches the native state directly, whereas the remaining fraction gets kinetically trapped in metastable conformations. For two-state folders Phi approximately 1. Molecular chaperones are recruited to assist protein folding whenever Phi is small. We show that the iterative annealing mechanism, introduced to describe chaperonin-mediated folding, can be generalized to understand protein-assisted RNA folding. The major differences between the folding of proteins and RNA arise in the early stages of folding. For RNA, folding can only begin after the polyelectrolyte problem is solved, whereas protein collapse requires burial of hydrophobic residues. Cross-fertilization of ideas between the two fields should lead to an understanding of how RNA and proteins solve their folding problems.

Large RNAs collapse into compact intermediates in the presence of counterions before folding to the native state. We previously found that collapse of a bacterial group I ribozyme correlates with the formation of helices within the ribozyme core, but occurs at Mg2+ concentrations too low to support stable tertiary structure and catalytic activity. Here, using small-angle X-ray scattering, we show that Mg2+-induced collapse is a cooperative folding transition that can be fit by a two-state model. The Mg2+ dependence of collapse is similar to the Mg2+ dependence of helix assembly measured by partial ribonuclease T1 digestion and of an unfolding transition measured by UV hypochromicity. The correspondence between multiple probes of RNA structure further supports a two-state model. A mutation that disrupts tertiary contacts between the L9 tetraloop and its helical receptor destabilized the compact state by 0.8 kcal/mol, while mutations in the central triplex were less destabilizing. These results show that native tertiary interactions stabilize the compact folding intermediates under conditions in which the RNA backbone remains accessible to solvent.

We calculate the mean end-to-end distance R of a self-avoiding polymer encapsulated in an infinitely long cylinder with radius D. A self-consistent perturbation theory is used to calculate R as a function of D for impenetrable hard walls and soft walls. In both cases, R obeys the predicted scaling behavior in the limit of large and small D. The crossover from the three-dimensional behavior (D –> infinity) to the fully stretched one-dimensional case (D –> 0) is nonmonotonic. The minimum value of R is found at D approximately 0.46R(F), where R(F) is the Flory radius of R at D –> infinity. The results for soft walls map onto the hard wall case with a larger cylinder radius.

We use a homotopy optimization method, HOPE, to minimize the potential energy associated with a protein model. The method uses the minimum energy conformation of one protein as a template to predict the lowest energy structure of a query sequence. This objective is achieved by following a path of conformations determined by a homotopy between the potential energy functions for the two proteins. Ensembles of solutions are produced by perturbing conformations along the path, increasing the likelihood of predicting correct structures. Successful results are presented for pairs of homologous proteins, where HOPE is compared to a variant of Newton’s method and to simulated annealing.

We develop a formalism for single molecule dynamic force spectroscopy to map the energy landscape of protein-protein complex (P(1)P(2)). The joint distribution P(tau(1),tau(2)) of unbinding lifetimes tau(1) and tau(2), measurable in a compression-tension cycle, which accounts for the internal relaxation dynamics of the proteins under tension, shows that the histogram of tau(1) is not Poissonian. The theory is applied to the forced unbinding of protein P1, modeled as a wormlike chain, from P(1)P(2). We propose a new class of experiments which can resolve the effect of internal protein dynamics on the unbinding lifetimes.

Several experiments have suggested that newly synthesized polypeptide chains can adopt helical structures deep within the ribosome exit tunnel. We hypothesize that confinement in the roughly cylindrical tunnel can entropically stabilize alpha-helices. The hypothesis is validated by using theory and simulations of coarse-grained off-lattice models. The model helix, which is unstable in the bulk, is stabilized in a cylindrical cavity provided the diameter (D) of the cylinder exceeds a critical value D*. When D < D* both the helical content and the helix-coil transition temperature (T(f)) decrease abruptly. Surprisingly, we find that the stability of the alpha-helix depends on the number (N) of amino acid residues. Entropic stabilization, as measured by changes in T(f), increases nonlinearly as N increases. The simulation results are in quantitative agreement with a standard helix-coil theory that takes into account entropy cost of confining a polypeptide chain in a cylinder. The results of this work are in qualitative accord with most of the findings of a recent experiment in which N-dependent ribosome-induced helix stabilization of transmembrane sequences was measured by fluorescence resonance energy transfer.

The influence of native connectivity of secondary structure elements (SSE) on folding is studied using coarse-grained models of proteins with mixed alpha and beta structure and the analysis of the structural database of wild-type proteins. We found that the distribution of SSE along a sequence determines the diversity of folding pathways. If alpha and beta SSE are localized in different parts of a sequence, the diversity of folding pathways is restricted. An even (symmetric) distribution of alpha and beta SSE with respect to sequence midpoint favors multiple folding routes. Simulations are supplemented by the database analysis of the distribution of SSE in wild-type protein sequences. On an average, two-thirds of wild-type proteins with mixed alpha and beta structure have symmetric distribution of alpha and beta SSE. The propensity for symmetric distribution of SSE is especially evident for large proteins with the number of SSE > or = 10. We suggest that symmetric SSE distribution in protein sequences may arise due to nearly random allocation of alpha and beta structure along wild-type sequences. The tendency of long sequences to misfold is perhaps compensated by the enhanced pathway diversity. In addition, folding pathways are shown to progress via hierarchic assembly of SSE in accordance with their proximity along a sequence. We demonstrate that under mild denaturation conditions folding and unfolding pathways are similar. However, the reversibility of folding/unfolding pathways is shown to depend on the distribution of SSE. If alpha and beta SSE are localized in different parts of a sequence, folding and unfolding pathways are likely to coincide.

We have investigated the folding and forced unbinding transitions of adsorbed semiflexible polymer chains using theory and simulations. These processes describe, at an elementary level, a number of biologically relevant phenomena that include adhesive interactions between proteins and tethering of receptors to cell walls. The binding interface is modeled as a solid surface, and the wormlike chain (WLC) is used for the semiflexible chain (SC). Using Langevin simulations, in the overdamped limit we examine the ordering kinetics of racquet-like and toroidal structures in the presence of an attractive interaction between the surface and the polymer chain. For a range of interactions, temperature, and the persistence length, l(p), we obtained the monomer density distribution, n(x), (x is the perpendicular distance of a tagged chain end from the surface) for all of the relevant morphologies. There is a single peak in n(x) inside the range of attractive forces, b, for chains in the extended conformations, whereas in racquet and toroidal structures there is an additional peak at x approximately b. The simulated results for n(x) are in good agreement with theory. The formation of toroids on the surface appears to be a first-order transition as evidenced by the bimodal distribution in n(x). The theoretical result underestimates the simulated n(x) for x < b and follows n(x) closely for x >/= b; the calculated density agrees exactly with n(x) in the range x < b. The chain-surface interaction is probed by subjecting the surface structures to a pulling force, f. The average extension, x( f), as a function of f exhibits a sigmoidal profile with sharp all-or-none transition at the unfolding force threshold f = f(c) which increases for more structured states. Simulated x(f) compare well with the theoretical predictions. The critical force, f(c), is a function of l(s)/l(c) for a fixed temperature, where l(c) and l(s) are the length scales that express the strength of the intramolecular and SC-surface attraction, respectively. For a fixed l(s), f(c) increases as l(p) decreases.

We determine the persistence length l(p) for a bacterial group I ribozyme as a function of concentration of monovalent and divalent cations by fitting the distance distribution functions P(r) obtained from small angle x-ray scattering intensity data to the asymptotic form of the calculated P(WLC)(r) for a wormlike chain. The l(p) values change dramatically over a narrow range of Mg(2+) concentration from approximately 21 Angstroms in the unfolded state (U) to approximately 10 Angstroms in the compact (I(C)) and native states. Variations in l(p) with increasing Na(+) concentration are more gradual. In accord with the predictions of polyelectrolyte theory we find l(p) alpha 1/kappa(2) where kappa is the inverse Debye-screening length.

We develop coarse-grained, distance- and orientation-dependent statistical potentials from the growing protein structural databases. For protein structural classes (alpha, beta, and alpha/beta), a substantial number of backbone-backbone and backbone-side-chain contacts stabilize the native folds. By taking into account the importance of backbone interactions with a virtual backbone interaction center as the 21st anisotropic site, we construct a 21 x 21 interaction scheme. The new potentials are studied using spherical harmonics analysis (SHA) and a smooth, continuous version is constructed using spherical harmonic synthesis (SHS). Our approach has the following advantages: (1) The smooth, continuous form of the resulting potentials is more realistic and presents significant advantages for computational simulations, and (2) with SHS, the potential values can be computed efficiently for arbitrary coordinates, requiring only the knowledge of a few spherical harmonic coefficients. The performance of the new orientation-dependent potentials was tested using a standard database of decoy structures. The results show that the ability of the new orientation-dependent potentials to recognize native protein folds from a set of decoy structures is strongly enhanced by the inclusion of anisotropic backbone interaction centers. The anisotropic potentials can be used to develop realistic coarse-grained simulations of proteins, with direct applications to protein design, folding, and aggregation.

We use long multiple trajectories generated by molecular dynamics simulations to probe the stability of oligomers of Abeta(16-22) (KLVFFAE) peptides in aqueous urea solution. High concentration of urea promotes the formation of beta-strand structures in Abeta(16-22) monomers, whereas in water they adopt largely compact random coil structures. The tripeptide system, which forms stable antiparallel beta-sheet structure in water, is destabilized in urea solution. The enhancement of beta-strand content in the monomers and the disruption of oligomeric structure occur largely by direct interaction of urea with the peptide backbone. Our simulations suggest that the oligomer unbinding dynamics is determined by two opposing effects, namely, by the increased propensity of monomers to form beta-strands and the rapid disruption of the oligomers. The qualitative conclusions are affirmed by using two urea models. Because the proposed destabilization mechanism depends largely on hydrogen bond formation between urea and the peptide backbone, we predict that high urea concentration will destabilize oligomers of other amyloidogenic peptides as well.

A new method is presented for extracting statistical potentials dependent on the relative side chain and backbone orientations in proteins. Coarse-grained, anisotropic potentials are constructed for short-, medium-, and long-range interactions using the Boltzmann method and a database of non-homologous protein structures. The new orientation-dependent potentials are analyzed using a spherical harmonics decomposition method with real eigenfunctions. This method permits a more realistic, continuous angular representation of the coarse-grained potentials. Results of tests for discriminating the native protein conformations from large sets of decoy proteins, show that the new continuous distance- and orientation-dependent potentials present significantly improved performance. Novel graphical representations are developed and used to depict the orientational dependence of the interaction potentials. These new continuous anisotropic statistical potentials could be instrumental in developing new computational methods for structure prediction, threading and coarse-grained simulations.

The need to perform large-scale studies of protein fold recognition, structure prediction and protein-protein interactions has led to novel developments of residue-level minimal models of proteins. A minimum requirement for useful protein force-fields is that they be successful in the recognition of native conformations. The balance between the level of detail in describing the specific interactions within proteins and the accuracy obtained using minimal protein models is the focus of many current protein studies. Recent results suggest that the introduction of explicit orientation dependence in a coarse-grained, residue-level model improves the ability of inter-residue potentials to recognize the native state. New statistical and optimization computational algorithms can be used to obtain accurate residue-dependent potentials for use in protein fold recognition and, more importantly, structure prediction.

Finite size effects on the cooperative thermal denaturation of proteins are considered. A dimensionless measure of cooperativity, Omegac, scales as Nzeta, where N is the number of amino acids. Surprisingly, we find that zeta is universal with zeta=1+gamma, where the exponent gamma characterizes the divergence of the susceptibility for a self-avoiding walk. Our lattice model simulations and experimental data are consistent with the theory. Our finding rationalizes the marginal stability of proteins and substantiates the earlier predictions that the efficient folding of two-state proteins requires TF approximately Ttheta, where Ttheta and TF are the collapse and folding transition temperatures, respectively.

Polyamines are abundant metabolites that directly influence gene expression. Although the role of polyamines in DNA condensation is well known, their role in RNA folding is less understood. Non-denaturing gel electrophoresis was used to monitor the equilibrium folding transitions of the Tetrahymena ribozyme in the presence of polyamines. All of the polyamines tested induce near-native structures that readily convert to the native conformation in Mg(2+). The stability of the folded structure increases with the charge of the polyamine and decreases with the size of the polyamine. When the counterion excluded volume becomes large, the transition to the native state does not go to completion even under favorable folding conditions. Brownian dynamics simulations of a model polyelectrolyte suggest that the kinetics of counterion-mediated collapse and the dimensions of the collapsed RNA chains depend on the structure of the counterion. The results are consistent with delocalized condensation of polyamines around the RNA. However, the effective charge of the counterions is lowered by their excluded volume. The stability of the folded RNA is enhanced when the spacing between amino groups matches the distance between adjacent phosphate groups. These results show how changes in intracellular polyamine concentrations could alter RNA folding pathways.

The first step in the formation of the protease resistant form (PrPSc) of prion proteins involves a conformational transition of the monomeric cellular form of PrPC to a more stable aggregation prone state PrPC*. A search of PDBselect and Escherichia coli and yeast genomes shows that the exact pattern of charges in helix 1 (H1) is rare. Among the 23 fragments in PDBselect with the pattern of charges that match H1, 83% are helical. Mapping of the rarely found (in E. coli and yeast genomes) hydrophobicity patterns in helix 2 (H2) to known secondary structures suggests that the PrPC–>PrPC* transition must be accompanied by alterations in conformations in second half of H2. We probe the dynamical instability in H1 and in the combined fragments of H2 and helix 3 (H3) from mPrPC (H2+H3), with intact disulfide bond, using all atom molecular dynamics (MD) simulations totaling 680 ns. In accord with recent experiments, we found that H1 is helical, whereas the double mutant H1[D147A-R151A] is less stable, implying that H1 is stabilized by the (i,i + 4) charged residues. The stability of H1 suggests that it is unlikely to be involved in the PrPC–>PrPC* transition. MD simulations of H2+H3 shows that the second half of H2 (residues 184-194) and parts of H3 (residues 200-204 and 215-223) undergo a transition from alpha-helical conformation to a beta and/or random coil state. Simulations using two force fields (optimized potentials for liquid simulations and CHARMM) give qualitatively similar results. We use the MD results to propose tentative structures for the PrPC* state.

MOTIVATION: Function of proteins or a network of interacting proteins often involves communication between residues that are well separated in sequence. The classic example is the participation of distant residues in allosteric regulation. Bioinformatic and structural analysis methods have been introduced to infer residues that are correlated. Recently, increasing attention has been paid to obtain the sequence properties that determine the tendency of disease-related proteins (Abeta peptides, prion proteins, transthyretin, etc.) to aggregate and form fibrils. Motivated in part by the need to identify sequence characteristics that indicate a tendency to aggregate, we introduce a general method that probes covariations in charged residues along the sequence in a given protein family. The method, which involves computing the sequence correlation entropy (SCE) using the quenched probability P(sk)(i,j) of finding a residue pair at a given sequence separation, sk, allows us to classify protein families in terms of their SCE. Our general approach may be a useful way in obtaining evolutionary covariations of amino acid residues on a genome wide level. RESULTS: We use a combination of SCE and clustering based on the principle component analysis to classify the protein families. From an analysis of 839 families, covering approximately 500,000 sequences, we find that proteins with relatively low values of SCE are predominantly associated with various diseases. In several families, residues that give rise to peaks in P(sk)(i,j) are clustered in the three-dimensional structure. For the class of proteins with low SCE values, there are significant numbers of mixed charged-hydrophobic (CH) and charged-polar (CP) runs. Our findings suggest that the low values of SCE and the presence of (CH) and/or (CP) may be indicative of disease association or tendency to aggregate. Our results led to the hypothesis that functions of proteins with similar SCE values may be linked. The hypothesis is validated with a few anecdotal examples. The present results also lead to the prediction that the overall charge correlations in proteins affect the kinetics of amyloid formation–a feature that is common to all proteins implicated in neurodegenerative diseases.

We present theory and simulations to describe nonequilibrium stretching of semiflexible chains that serve as models of DNA molecules. Using a self-consistent dynamical variational approach, we calculate the force-extension curves for worm-like chains as a function of the pulling speed, v(0). Due to nonequilibrium effects the stretching force, which increases with v(0), shows nonmonotonic variations as the persistence length increases. To complement the theoretical calculations we also present Langevin simulation results for extensible worm-like chain models for the dynamics of stretching. The theoretical force-extension predictions compare well with the simulation results. The simulations show that, at high enough pulling speeds, the propagation of tension along the chain conformations transverse to the applied force occurs by the Brochard-Wyart’s stem-flower mechanism. The predicted nonequilibrium effects can only be observed in double-stranded DNA at large ( approximately 100 microm/s) pulling speeds.

The Escherichia coli chaperonin system, GroEL-GroES, facilitates folding of substrate proteins (SPs) that are otherwise destined to aggregate. The iterative annealing mechanism suggests that the allostery-driven GroEL transitions leading to changes in the microenvironment of the SP constitutes the annealing action of chaperonins. To describe the molecular basis for the changes in the nature of SP-GroEL interactions we use the crystal structures of GroEL (T state), GroEL-ATP (R state) and the GroEL-GroES-(ADP)(7) (R” state) complex to determine the residue-specific changes in the accessible surface area and the number of tertiary contacts as a result of the T–>R–>R” transitions. We find large changes in the accessible area in many residues in the apical domain, but relatively smaller changes are associated with residues in the equatorial domain. In the course of the T–>R transition the microenvironment of the SP changes which suggests that GroEL is an annealing machine even without GroES. This is reflected in the exposure of Glu386 which loses six contacts in the T–>R transition. We also evaluate the conservation of residues that participate in the various chaperonin functions. Multiple sequence alignments and chemical sequence entropy calculations reveal that, to a large extent, only the chemical identities and not the residues themselves important for the nominal functions (peptide binding, nucleotide binding, GroES and substrate protein release) are strongly conserved. Using chemical sequence entropy, which is computed by classifying aminoacids into four types (hydrophobic, polar, positively charged and negatively charged) we make several new predictions that are relevant for peptide binding and annealing function of GroEL. We identify a number of conserved peptide binding sites in the apical domain which coincide with those found in the 1.7 A crystal structure of ‘mini-chaperone’ complexed with the N-terminal tag. Correlated mutations in the HSP60 family, that might control allostery in GroEL, are also strongly conserved. Most importantly, we find that charged solvent-exposed residues in the T state (Lys 226, Glu 252 and Asp 253) are strongly conserved. This leads to the prediction that mutating these residues, that control the annealing function of the SP, can decrease the efficacy of the chaperonin function.

By considering temperature effects on the mechanical unfolding rates of proteins and RNA, whose energy landscape is rugged, the question posed in the title is answered in the affirmative. Adopting a theory by Zwanzig [Zwanzig, R. (1988) Proc. Natl. Acad. Sci. USA 85, 2029-2030], we show that, because of roughness characterized by an energy scale epsilon, the unfolding rate at constant force is retarded. Similarly, in nonequilibrium experiments done at constant loading rates, the most probable unfolding force increases because of energy landscape roughness. The effects are dramatic at low temperatures. Our analysis suggests that, by using temperature as a variable in mechanical unfolding experiments of proteins and RNA, the ruggedness energy scale epsilon, can be directly measured.

Multiple long molecular dynamics simulations are used to probe the oligomerization mechanism of Abeta(16-22) (KLVFFAE) peptides. The peptides, in the monomeric form, adopt either compact random-coil or extended beta strand-like structures. The assembly of the low-energy oligomers, in which the peptides form antiparallel beta sheets, occurs by multiple pathways with the formation of an obligatory alpha-helical intermediate. This observation and the experimental results on fibrillogenesis of Abeta(1-40) and Abeta(1-42) peptides suggest that the assembly mechanism (random coil –> alpha helix –> beta strand) is universal for this class of peptides. In Abeta(16-22) oligomers both interpeptide hydrophobic and electrostatic interactions are critical in the formation of the antiparallel beta sheet structure. Mutations of either hydrophobic or charged residues destabilize the oligomer, which implies that the 16-22 fragments of Arctic (E22G), Dutch (E22Q), and Italian (E22K) mutants are unlikely to form ordered fibrils.

The conformational equilibrium of a blocked valine peptide in water and aqueous urea solution is studied using molecular dynamics simulations. Pair correlation functions indicate enhanced concentration of urea near the peptide. Stronger hydrogen bonding of urea-peptide compared to water-peptide is observed with preference for helical conformation. The potential of mean force, computed using umbrella sampling, shows only small differences between urea and water solvation that are difficult to quantify. The changes in solvent structure around the peptide are explained by favorable electrostatic interactions (hydrogen bonds) of urea with the peptide backbone. There is no evidence for significant changes in hydrophobic interactions in the two conformations of the peptide in urea solution. Our simulations suggest that urea denatures proteins by preferentially forming hydrogen bonds to the peptide backbone, reducing the barrier for exposing protein residues to the solvent, and reaching the unfolded state.

Several neurodegenerative diseases are associated with the unfolding and subsequent fibrillization of proteins. Although neither the assembly mechanism nor the atomic structures of the amyloid fibrils are known, recent experimental and computational studies suggest that a few general principles that govern protein aggregation may exist. Analysis of the results of several important recent studies has led to a set of tentative ideas concerning the oligomerization of proteins and peptides. General rules have been described that may be useful in predicting regions of known proteins (prions and transthyretin) that are susceptible to fluctuations, which give rise to structures that can aggregate by the nucleation-growth mechanism. Despite large variations in the sequence-dependent polymerization kinetics of several structurally unrelated proteins, there appear to be only a few plausible scenarios for protein and peptide aggregation.

We probe the urea-denaturation mechanism using molecular dynamics simulations of an elementary “folding” event, namely, the formation of end-to-end contact in the linear hydrocarbon chain (HC) CH(3)(CH(2))(18)CH(3). Electrostatic effects are examined using a model HC in which one end of the chain is positively charged (+0.2e) and the other contains a negative charge (-0.2e). For these systems multiple transitions between “folded” (conformations in which the chain ends are in contact) and “unfolded” (end-to-end contact is broken) can be observed during 4 ns molecular dynamics simulations. In water and 6 M aqueous urea solution HC and the charged HC fluctuate between collapsed globular conformations and a set of expanded structures. The collapsed conformation adopted by the HC in water is slightly destablized in 6 M urea. In contrast, the end-to-end contact is disrupted in the charged HC only in aqueous urea solution. Despite the presence of a large hydrophobic patch, on length scales on the order of approximately 8-10 A “denaturation” (transition to the expanded unfolded state) occurs by a direct interaction of urea with charges on the chain ends. The contiguous patch of hydrophobic moieties leads to “mild dewetting”, which becomes more pronounced in the charged HC in 6 M aqueous urea solution. Our simulations establish that the urea denaturation mechanism is most likely electrostatic in origin.

The activity of the Alzheimer’s amyloid beta-peptide is a sensitive function of the peptide’s sequence. Increased fibril elongation rate of the E22Q Dutch mutant of the Alzheimer’s amyloid beta-peptide relative to that of the wild-type peptide has been observed. The increased activity has been attributed to a larger propensity for the formation of beta structure in the monomeric E22Q mutant peptide in solution relative to the WT peptide. That hypothesis is tested using four nanosecond timescale simulations of the WT and Dutch mutant forms of the Abeta(10-35)-peptide in aqueous solution. The simulation results indicate that the propensity for formation of beta-structure is no greater in the E22Q mutant peptide than in the WT peptide. A significant measure of “flickering” of helical structure in the central hydrophobic cluster region of both the WT and mutant peptides is observed. The simulation results argue against the hypothesis that the Dutch mutation leads to a higher probability of formation of beta-structure in the monomeric peptide in aqueous solution. We propose that the greater stability of the solvated WT peptide relative to the E22Q mutant peptide leads to decreased fibril elongation rate in the former. Stability difference is due to the differing charge state of the two peptides. The other proposal leads to the prediction that the fibril elongation rates for the WT and the mutant E22Q should be similar under acid conditions.

Many seemingly unrelated neurodegenerative disorders, such as amyloid and prion diseases, are associated with propagating fibrils whose structures are dramatically different from the native states of the corresponding monomers. This observation, along with the experimental demonstration that any protein can aggregate to form either fibrils or amorphous structures (inclusion bodies) under appropriate external conditions, suggest that there must be general principles that govern aggregation mechanisms. To probe generic aspects of prion-like behavior we use the model of Harrison, Chan, Prusiner, and Cohen. In this model, aggregation of a structure, that is conformationally distinct from the native state of the monomer, occurs by three parallel routes. Kinetic partitioning, which leads to parallel assembly pathways, occurs early in the aggregation process. In all pathways transient unfolding precedes polymerization and self-propagation. Chain polymerization is consistent with templated assembly, with the dimer being the minimal nucleus. The kinetic effciency of R(n-1) + G –> R(n) (R is the aggregation prone state and G is either U, the unfolded state, or N, the native state of the monomer) is increased when polymerization occurs in the presence of a “seed” (a dimer). These results support the seeded nucleated-polymerization model of fibril formation in amyloid peptides. To probe generic aspects of aggregation in two-state proteins, we use lattice models with side chains. The phase diagram in the (T,C) plane (T is the temperature and C is the polypeptide concentration) reveals a bewildering array of “phases” or structures. Explicit computations for dimers show that there are at least six phases including ordered structures and amorphous aggregates. In the ordered region of the phase diagram there are three distinct structures. We find ordered dimers (OD) in which each monomer is in the folded state and the interaction between the monomers occurs via a well-defined interface. In the domain-swapped structures a certain fraction of intrachain contacts are replaced by interchain contacts. In the parallel dimers the interface is stabilized by favorable intermolecular hydrophobic interactions. The kinetics of folding to OD shows that aggregation proceeds directly from U in a dynamically cooperative manner without populating partially structured intermediates. These results support the experimental observation that ordered aggregation in the two-state folders U1A and CI2 takes place from U. The contrasting aggregation processes in the two models suggest that there are several distinct mechanisms for polymerization that depend not only on the polypeptide sequence but also on external conditions (such as C, T, pH, and salt concentration).

We report the thermodynamics and kinetics of an off-lattice Go model beta-hairpin from Ig-binding protein confined to an inert spherical pore. Confinement enhances the stability of the hairpin due to the decrease in the entropy of the unfolded state. Compared with their values in the bulk, the rates of hairpin formation increase in the spherical pore. Surprisingly, the dependence of the rates on the pore radius, R(s), is nonmonotonic. The rates reach a maximum at R(s)/R(g,N)(b) approximately equal to 1.5, where R(g,N)(b) is the radius of gyration of the folded beta-hairpin in the bulk. The denatured state ensemble of the encapsulated beta-hairpin is highly structured even at substantially elevated temperatures. Remarkably, a profound effect of confinement is evident even when the beta-hairpin occupies less than a 10th of the sphere volume. Our calculations show that the emergence of substantial structure in the denatured state of proteins in inert pores is a consequence of confinement. In contrast, the structure of the bulk denatured state ensemble depends dramatically on the extent of denaturation.

Protein engineering experiments and Phi(F)-value analysis of SH3 domains reveal that their transition state ensemble (TSE) is conformationally restricted, i.e. the fluctuations in the transition state (TS) structures are small. In the TS of src SH3 and alpha-spectrin SH3 the distal loop and the associated hairpin are fully structured, while the rest of the protein is relatively disordered. If native structure predominantly determines the folding mechanism, the findings for SH3 folds raise the question: What are the features of the native topology that determine the nature of the TSE? We propose that the presence of stiff loops in the native state that connect local structural elements (such as the distal hairpin in SH3 domains) conformationally restricts TSE. We validate this hypothesis using the simulations of a “control” system (16 residue beta-hairpin forming C-terminal fragment of the GBl protein) and its variants. In these fragments the role of bending rigidity in determining the nature of the TSE can be directly examined without complications arising from interactions with the rest of the protein. The TSE structures in the beta-hairpins are determined computationally using cluster analysis and limited Phi(F)-value analysis. Both techniques prove that the conformational heterogeneity decreases as the bending rigidity of the loop increases. To extend this finding to SH3 domains a measure of bending rigidity based on loop curvature, which utilizes native structures in the Protein Data Bank (PDB), is introduced. Using this measure we show that, with few exceptions, the ordering of stiffness of the distal, n-src, and RT loops in the 29 PDB structures of SH3 domains is conserved. Combining the simulation results for beta-hairpins and the analysis of PDB structures for SH3 domains, we propose that the stiff distal loop restricts the conformational fluctuations in the TSE. We also predict that constraining the distal loop to be preformed in the denatured ensemble should not alter the nature of TSE. On the other hand, if the amino and carboxy terminals are cross-linked to form a circular polypeptide chain, the pathways and TSs are altered. These contrasting scenarios are illustrated using simulations of cross-linked WT beta-hairpin fragments. Computations of bending rigidities for immunoglobulin-like domain proteins reveal no clear separation in the stiffness of their loops. In the beta-sandwich proteins, which have large fractions of non-local native contacts, the nature of the TSE cannot be apparently determined using purely local structural characteristics. Nevertheless, the measure of loop stiffness still provides qualitative predictions of the ordered regions in the TSE of Ig27 and TenFn3.

Thermal unfolding (or folding) in many proteins occurs in an apparent two-state manner, suggesting that only two states, unfolded and folded, are populated. At the melting temperature, Tm, the two states coexist. Using lattice models with side chains we show that individual residues become structured at temperatures that deviate from Tm, which implies that partially folded conformations make substantial contribution to thermodynamic properties of two-state proteins. We also find that the folding cooperativity for a given residue is linked to its accessible surface area. These results are consistent with the experiments on GCN4-like zipper peptide, which showed that local melting temperatures differ from Tm. Analysis of thermal unfolding of six proteins shows that deltaT/Tm approximately N(-1), where deltaT is the transition width and N is the number of residues. This scaling allows us to conclude that, when corrected for finite size effects, folding cooperativity can be captured using coarse grained models.

Neurodegenerative diseases induced by transmissible spongiform encephalopathies are associated with prions. The most spectacular event in the formation of the infectious scrapie form, referred to as PrP(Sc), is the conformational change from the predominantly alpha-helical conformation of PrP(C) to the PrP(Sc) state that is rich in beta-sheet content. Using sequence alignments and structural analysis of the available nuclear magnetic resonance structures of PrP(C), we explore the propensities of helices in PrP(C) to be in a beta-strand conformation. Comparison of a number of structural characteristics (such as solvent accessible area, distribution of (Phi, Psi) angles, mismatches in hydrogen bonds, nature of residues in local and nonlocal contacts, distribution of regular densities of amino acids, clustering of hydrophobic and hydrophilic residues in helices) between PrP(C) structures and a databank of “normal” proteins shows that the most unusual features are found in helix 2 (H2) (residues 172-194) followed by helix 1 (H1) (residues 144-153). In particular, the C-terminal residues in H2 are frustrated in their helical state. The databank of normal proteins consists of 58 helical proteins, 36 alpha+beta proteins, and 31 beta-sheet proteins. Our conclusions are also substantiated by gapless threading calculations that show that the normalized Z-scores of prion proteins are similar to those of other alpha+beta proteins with low helical content. Application of the recently introduced notion of discordance, namely, incompatibility of the predicted and observed secondary structures, also points to the frustration of H2 not only in the wild type but also in mutants of human PrP(C). This suggests that the instability of PrP(C) proteins may play a role in their being susceptible to the profound conformational change. Our analysis shows that, in addition to the previously proposed role for the segment (90-120) and possibly H1, the C-terminus of H2 and possibly N-terminus may play a role in the alpha–>beta transition. An implication of our results is that the ease of polymerization depends on the unfolding rate of the monomer. Sequence alignments show that helices in avian prion proteins (chicken, duck, crane) are better accommodated in a helical state, which might explain the absence of PrP(Sc) formation over finite time scales in these species. From this analysis, we predict that correlated mutations that reduce the frustration in the second half of helix 2 in mammalian prion proteins could inhibit the formation of PrP(Sc).

Folding of RNA into an ordered, compact structure requires substantial neutralization of the negatively charged backbone by positively charged counterions. Using a native gel electrophoresis assay, we have examined the effects of counterion condensation upon the equilibrium folding of the Tetrahymena ribozyme. Incubation of the ribozyme in the presence of mono-, di- and trivalent ions induces a conformational state that is capable of rapidly forming the native structure upon brief exposure to Mg2+. The cation concentration dependence of this transition is directly correlated with the charge of the counterion used to induce folding. Substrate cleavage assays confirm the rapid onset of catalytic activity under these conditions. These results are discussed in terms of classical counterion condensation theory. A model for folding is proposed which predicts effects of charge, ionic radius and temperature on counterion-induced RNA folding transitions.

Molecular chaperones are required to assist folding of a subset of proteins in Escherichia coli. We describe a conceptual framework for understanding how the GroEL-GroES system assists misfolded proteins to reach their native states. The architecture of GroEL consists of double toroids stacked back-to-back. However, most of the fundamentals of the GroEL action can be described in terms of the single ring. A key idea in our framework is that, with coordinated ATP hydrolysis and GroES binding, GroEL participates actively by repeatedly unfolding the substrate protein (SP), provided that it is trapped in one of the misfolded states. We conjecture that the unfolding of SP becomes possible because a stretching force is transmitted to the SP when the GroEL particle undergoes allosteric transitions. Force-induced unfolding of the SP puts it on a higher free-energy point in the multidimensional energy landscape from which the SP can either reach the native conformation with some probability or be trapped in one of the competing basins of attraction (i.e., the SP undergoes kinetic partitioning). The model shows, in a natural way, that the time scales in the dynamics of the allosteric transitions are intimately coupled to folding rates of the SP. Several scenarios for chaperonin-assisted folding emerge depending on the interplay of the time scales governing the cycle. Further refinement of this framework may be necessary because single molecule experiments indicate that there is a great dispersion in the time scales governing the dynamics of the chaperonin cycle.

We describe a conceptual framework for understanding the way large RNA molecules fold based on the notion that their free-energy landscape is rugged. A key prediction of our theory is that RNA folding can be described by the kinetic partitioning mechanism (KPM). According to KPM a small fraction of molecules folds rapidly to the native state whereas the remaining fraction is kinetically trapped in a low free-energy non-native state. This model provides a unified description of the way RNA and proteins fold. Single-molecule experiments on Tetrahymena ribozyme, which directly validate our theory, are analyzed using KPM. We also describe the earliest events that occur on microsecond time scales in RNA folding. These must involve collapse of RNA molecules that are mediated by counterion-condensation. Estimates of time scales for the initial events in RNA folding are provided for the Tetrahymena ribozyme.

Using exhaustive simulations of lattice models with side-chains, we show that optimized two-state folders reach the native state by a nucleation-collapse mechanism with multiple folding nuclei (MFN). For both the full model and the Go version, there are certain contacts that on an average participate in the critical nuclei with higher probability than the others. The high- (> or = 0.5) probability contacts are largely determined by the structure of the native state. Comparison of the results for the full sequence and the Go model shows that non-native interactions compromise the degree of cooperativity and stability of the native state. From an extremely detailed analysis of the folding kinetics, we find that non-native interactions are present in the folding nuclei. The folding times decrease if the non-native interactions in the folding nuclei are made neutral or repulsive. Using cluster analysis and making no prior assumption about reaction coordinate, we show that both full and Go models have three distinct transition states that give a structural description for the MFN. In the transition states, on an average, about two-thirds of the sequence is structured, whereas the rest is disordered, reminiscent of the polarized transition state in the SH3 domain. Our studies show that Go models cannot describe the transition state characteristics of two-state folders at the molecular level. As a byproduct of our investigations, we establish that our method of computing the transition state ensemble is numerically equivalent to the technique based on the stochastic separatrix, which also does not require a priori knowledge of the folding reaction coordinate.

Condensed counterions contribute to the stability of compact structures in RNA, largely by reducing electrostatic repulsion among phosphate groups. Varieties of cations induce a collapsed state in the Tetrahymena ribozyme that is readily transformed to the catalytically active structure in the presence of Mg2+. Native gel electrophoresis was used to compare the effects of the valence and size of the counterion on the kinetics of this transition. The rate of folding was found to decrease with the charge of the counterion. Transitions in monovalent ions occur 20- to 40-fold faster than transitions induced by multivalent metal ions. These results suggest that multivalent cations yield stable compact structures, which are slower to reorganize to the native conformation than those induced by monovalent ions. The folding kinetics are 12-fold faster in the presence of spermidine3+ than [Co(NH3)6]3+, consistent with less effective stabilization of long-range RNA interactions by polyamines. Under most conditions, the observed folding rate decreases with increasing counterion concentration. In saturating amounts of counterion, folding is accelerated by addition of urea. These observations indicate that reorganization of compact intermediates involves partial unfolding of the RNA. We find that folding of the ribozyme is most efficient in a mixture of monovalent salt and Mg2+. This is attributed to competition among counterions for binding to the RNA. The counterion dependence of the folding kinetics is discussed in terms of the ability of condensed ions to stabilize compact structures in RNA.

Large ribozymes typically require very long times to refold into their active conformation in vitro, because the RNA is easily trapped in metastable misfolded structures. Theoretical models show that the probability of misfolding is reduced when local and long-range interactions in the RNA are balanced. Using the folding kinetics of the Tetrahymena ribozyme as an example, we propose that folding rates are maximized when the free energies of forming independent domains are similar to each other. A prediction is that the folding pathway of the ribozyme can be reversed by inverting the relative stability of the tertiary domains. This result suggests strategies for optimizing ribozyme sequences for therapeutics and structural studies.

Thermodynamics and kinetics of off-lattice models with side chains for the beta-hairpin fragment of immunoglobulin-binding protein and its variants are reported. For all properties (except refolding time tau(F)) there are no qualitative differences between the full model and the Go version. The validity of the models is established by comparison of the calculated native structure with the Protein Data Bank coordinates and by reproducing the experimental results for the degree of cooperativity and tau(F). For the full model tau(F) approximately 2 micros at the folding temperature (experimental value is 6 micros); the Go model folds 50 times faster. Upon refolding, structural changes take place over three time scales. On the collapse time scale compact structures with intact hydrophobic cluster form. Subsequently, hydrogen bonds form, predominantly originating from the turn by a kinetic zipping mechanism. The assembly of the hairpin is complete when most of the interstrand contacts (the rate-limiting step) is formed. The dominant transition state structure (located by using cluster analysis) is compact and structured. We predict that when hydrophobic cluster is moved to the loop tau(F) marginally increases, whereas moving the hydrophobic cluster closer to the termini results in significant decrease in tau(F) relative to wild type. The mechanism of hairpin formation is predicted to depend on turn stiffness.

Single-molecule manipulation techniques reveal that stretching unravels individually folded domains in the muscle protein titin and the extracellular matrix protein tenascin. These elastic proteins contain tandem repeats of folded domains with beta-sandwich architecture. Herein, we propose by stretching two model sequences (S1 and S2) with four-stranded beta-barrel topology that unfolding forces and pathways in folded domains can be predicted by using only the structure of the native state. Thermal refolding of S1 and S2 in the absence of force proceeds in an all-or-none fashion. In contrast, phase diagrams in the force-temperature (f,T) plane and steered Langevin dynamics studies of these sequences, which differ in the native registry of the strands, show that S1 unfolds in an allor-none fashion, whereas unfolding of S2 occurs via an obligatory intermediate. Force-induced unfolding is determined by the native topology. After proving that the simulation results for S1 and S2 can be calculated by using native topology alone, we predict the order of unfolding events in Ig domain (Ig27) and two fibronectin III type domains ((9)FnIII and (10)FnIII). The calculated unfolding pathways for these proteins, the location of the transition states, and the pulling speed dependence of the unfolding forces reflect the differences in the way the strands are arranged in the native states. We also predict the mechanisms of force-induced unfolding of the coiled-coil spectrin (a three-helix bundle protein) for all 20 structures deposited in the Protein Data Bank. Our approach suggests a natural way to measure the phase diagram in the (f,C) plane, where C is the concentration of denaturants.

Considerable insights into the mechanisms and timescales of protein folding have been obtained from detailed studies of minimal off-lattice models. These models are coarse-grained representations of polypeptide chains. Many novel predictions of the mechanisms and timescales of the folding of proteins have been made using simulations of off-lattice models. The concepts derived from these simulations have been used to analyze the recent experiments and simulations of proteins and peptides.

The chaperonin system, GroEL and GroES of Escherichia coli enable certain proteins to fold under conditions when spontaneous folding is prohibitively slow as to compete with other non-productive channels such as aggregation. We investigated the plausible mechanisms of GroEL-mediated folding using simple lattice models. In particular, we have investigated protein folding in a confined environment, such as those offered by the GroEL, to decipher whether rate and yield enhancement can occur when the substrate protein is allowed to fold within the cavity of the chaperonins. The GroEL cavity is modeled as a cubic box and a simple bead model is used to represent the substrate chain. We consider three distinct characteristic of the confining environment. First, the cavity is taken to be a passive Anfinsen cage in which the walls merely reduce the available conformation space. We find that at temperatures when the native conformation is stable, the folding rate is retarded in the Anfinsen cage. We then assumed that the interior of the wall is hydrophobic. In this case the folding times exhibit a complex behavior. When the strength of the interaction between the polypeptide chain and the cavity is too strong or too weak we find that the rates of folding are retarded compared to spontaneous folding. There is an optimum range of the interaction strength that enhances the rates. Thus, above this value there is an inverse correlation between the folding rates and the strength of the substrate-cavity interactions. The optimal hydrophobic walls essentially pull the kinetically trapped states which leads to a smoother the energy landscape. It is known that upon addition of ATP and GroES the interior cavity of GroEL offers a hydrophilic-like environment to the substrate protein. In order to mimic this within the context of the dynamic Anfinsen cage model, we allow for changes in the hydrophobicity of the walls of the cavity. The duration for which the walls remain hydrophobic during one cycle of ATP hydrolysis is allowed to vary. These calculations show that frequent cycling of the wall hydrophobicity can dramatically reduce the folding times and increase the yield as well under non-permissive conditions. Examination of the structures of the substrate proteins before and after the change in hydrophobicity indicates that there is global unfolding involved. In addition, it is found that a fraction of the molecules kinetically partition to the native state in accordabce with the iterative annealing mechanism. Thus, frequent “unfoldase” activity of chaperonins leading to global unfolding of the polypeptide chain results in enhancement of the folding rates and yield of the folded protein. We suggest that chaperonin efficiency can be greatly enhanced if the cycling time is reduced. The calculations are used to interpret a few experiments on chaperonin-mediated protein folding.

Folding of the Tetrahymena self-splicing RNA into its active conformation involves a set of discrete intermediate states. The Mg2+-dependent equilibrium transition from the intermediates to the native structure is more cooperative than the formation of the intermediates from the unfolded states. We show that the degree of cooperativity is linked to the free energy of each transition and that the rate of the slow transition from the intermediates to the native state decreases exponentially with increasing Mg2+ concentration. Monovalent salts, which stabilize the folded RNA nonspecifically, induce states that fold in less than 30 s after Mg2+ is added to the RNA. A simple model is proposed that predicts the folding kinetics from the Mg2+-dependent change in the relative stabilities of the intermediate and native states.

We examine the similarities and differences between two widely used knowledge-based potentials, which are expressed as contact matrices (consisting of 210 elements) that gives a scale for interaction energies between the naturally occurring amino acid residues. These are the Miyazawa-Jernigan contact interaction matrix M and the potential matrix S derived by Skolnick J et al., 1997, Protein Sci 6:676-688. Although the correlation between the two matrices is good, there is a relatively large dispersion between the elements. We show that when Thr is chosen as a reference solvent within the Miyazawa and Jernigan scheme, the dispersion between the M and S matrices is reduced. The resulting interaction matrix B gives hydrophobicities that are in very good agreement with experiment. The small dispersion between the S and B matrices, which arises due to differing reference states, is shown to have dramatic effect on the predicted native states of lattice models of proteins. These findings and other arguments are used to suggest that for reliable predictions of protein structures, pairwise additive potentials are not sufficient. We also establish that optimized protein sequences can tolerate relatively large random errors in the pair potentials. We conjecture that three body interaction may be needed to predict the folds of proteins in a reliable manner.

Single-molecule force spectroscopy reveals unfolding of domains in titin on stretching. We provide a theoretical framework for these experiments by computing the phase diagrams for force-induced unfolding of single-domain proteins using lattice models. The results show that two-state folders (at zero force) unravel cooperatively, whereas stretching of non-two-state folders occurs through intermediates. The stretching rates of individual molecules show great variations reflecting the heterogeneity of force-induced unfolding pathways. The approach to the stretched state occurs in a stepwise “quantized” manner. Unfolding dynamics and forces required to stretch proteins depend sensitively on topology. The unfolding rates increase exponentially with force f till an optimum value, which is determined by the barrier to unfolding when f = 0. A mapping of these results to proteins shows qualitative agreement with force-induced unfolding of Ig-like domains in titin. We show that single-molecule force spectroscopy can be used to map the folding free energy landscape of proteins in the absence of denaturants.

BACKGROUND: Over the past few years novel folding mechanisms of globular proteins have been proposed using minimal lattice and off-lattice models. The factors determining the cooperativity of folding in these models and especially their explicit relation to experiments have not been fully established, however. RESULTS: We consider equilibrium folding transitions in lattice models with and without sidechains. A dimensionless measure, omega c, is introduced to quantitatively assess the degree of cooperativity in lattice models and in real proteins. We show that larger values of omega c resembling the values seen in proteins are obtained in lattice models with sidechains. The enhanced cooperativity of such models results from possible denser packing of sidechains in the interior of the model polypeptide chain. We also establish that omega c correlates extremely well with sigma T = (T o – T f) /T o, where T o and T f are collapse and folding transition temperatures, respectively. These theoretical ideas are used to analyze folding transitions in two-state folders (RNase A, chymotrypsin inhibitor 2, fibronectin type III modules and tendamistat) and three-state folders (apomyoglobin and lysozyme). The values of omega c extracted from experiments show a correlation with sigma T (suitably generalized when folding is induced by denaturants or acid). CONCLUSIONS: A quantitative description of the cooperative transition of real proteins can be made by lattice models with sidechains. The degree of cooperativity in minimal models and real proteins can be expressed in terms of the single parameter sigma, which can be estimated from experimental data.

The hydrophobic hydration in a series of hydrocarbons is probed by using molecular dynamics simulations. The solutes considered range from methane to octane. Examination of the shapes of the hydration shell suggests that there is no single stable structure surrounding these solutes. The structure of the water molecules around the solute is not significantly perturbed, even for octane, and the hydrogen bond network is essentially preserved. The solutes are accommodated in the voids of the tetrahedral network of water in such a way as to leave the local environment almost intact. The hydrophobic hydration arises primarily because of the plasticity of the hydrogen bond network. Even for octane we find very little evidence for water-mediated interactions between nonbonded carbon atoms, leading us to suggest that the transition to globular conformations can only occur for very long, linear hydrocarbon chains.

The nature of the nucleation-collapse mechanism in protein folding is probed using 27-mer and 36-mer lattice models. Three different forms for the interaction potentials are used. Three of the four 27-mer sequences have maximally compact and identical native state while the other has a non-compact native conformation. All the sequences fold thermodynamically and kinetically by a two-state process. Analysis of individual trajectories for each sequence using a self-organizing neural net algorithm shows that upon formation of a critical set of contacts the polypeptide chain rapidly reaches the native conformation which is consistent with a nucleation-collapse mechanism. The algorithm, which reduces the identification of the folding nucleus for each trajectory to one of pattern recognition, is used to show that there are multiple folding nuclei. There is a distribution of nucleation contacts in the transition states with some of them occurring with more probability (when averaged over the denatured ensemble) than others. We also show that there is a distribution in the size of the nuclei with the average number of residues in the folding nuclei being less than about one-third of the chain size. The fluctuations in the sizes of the nuclei are large, suggestive of a broad transition region. The folding nuclei, the structures of each are the corresponding transition states, have varying degree of overlap with the native conformation. The distribution of the radius of gyration of the transition states shows that these structures are an expanded form (by about 25% in the radius of gyration) of the native conformation. Local contacts are most dominant in the folding nuclei while a certain fraction of non-local contacts is necessary to stabilize the transition states. The search for the critical nuclei initially involves the formation of local contacts, while non-local contacts are formed later. The fractional values of PhiF for the two 27-mer mutants found by using the protein engineering protocol are consistent with the microscopic picture of partial formation of structures involving these residues in the transition state. These observations lead to a multiple folding nuclei (MFN) model for nucleation-collapse mechanism in protein folding. The major implication of the MFN model is that, even if the residues whose tertiary interactions are formed nearly completely in the transition state are mutated, it does not disrupt the nature of the nucleation-collapse mechanism. We analyze the experiments on chymotrypsin inhibitor 2 and alpha-spectrin SH3 domain and two circular permutants in light of the MFN model. It is shown that the PhiF-value analysis for these proteins gives considerable support to the MFN model. The theoretical and experimental studies give a coherent picture of the nucleation-collapse mechanism in which there is a distribution of folding nuclei with some more probable than others. The formation of any specific nucleus is not necessary for efficient two-state folding.

BACKGROUND: The most conspicuous feature of a right-handed alpha helix is the presence of hydrogen bonds between the backbone carbonyl oxygen and NH groups along the chain. A simple off-lattice model that includes hydrogen bond interactions using virtual atoms is used to examine the stability, cooperativity and kinetics of the helix-coil transition. RESULTS: We have studied the thermodynamics (using multiple histogram method) and kinetics (by Brownian dynamics simulations) of 16-mer minimal off-lattice models of four-turn alpha-helix sequences. The carbonyl and NH groups are represented as virtual moieties located between two alpha-carbon atoms along the polypeptide chain. The characteristics of the native conformations of the model helices, such as the helical pitch and angular correlations, coincide with those found in real proteins. The transition from coil to helix is quite broad, which is typical of these finite-sized systems. The cooperativity, as measured by a dimensionless parameter, omegac, that takes into account the width and the slope of the transition curves, is enhanced when hydrogen bonds are taken into account. The value of omegac for our model is consistent with that inferred from experiment for an alanine-based helix-forming peptide. The folding time tauF ranges from 6 to 1000 ns in the temperature range 0.7-1.9 T(F), where T(F) is the helix-coil transition temperature. These values are in excellent agreement with the results from recent fast folding experiments. The temperature dependence of tauF exhibits a nearly Arrhenius behavior. Thermally induced unfolding occurs on a time scale that is less than 40-170 ps depending on the final temperature. Our calculations also predict that, although tauF can be altered by changes in the sequence, the dynamic range over which such changes take place is not as large as that predicted for beta-turn formation. CONCLUSIONS: Hydrogen bonds not only affect the stability of alpha-helix formation but also have profound influence on the kinetics. The excellent agreement between our calculations and experiments suggests that these models can be used to investigate the effects of sequence, temperature and viscosity on the helix-coil transition.

Folding kinetics of large RNAs are just beginning to be investigated. We show that the Tetrahymena self-splicing RNA partitions into a population that rapidly reaches the native state, and a slowly folding population that is trapped in metastable misfolded structures. Transitions from the misfolded structures to the native state involve partial unfolding. The total yield of native RNA is increased by iterative annealing of the inactive population, and mildly denaturing conditions increase the rate of folding at physiological temperatures. These results provide the first evidence that an RNA can fold by multiple parallel paths.

The folding of Ser-Tyr-Pro-Phe-Asp-Val (SYPFDV), and sequence variants of this peptide (SYPYD and SYPFD) are studied computationally in an explicit water environment. An atomically detailed model of the peptide is embedded in a sphere of TIP3P water molecules and its optimal structure is computed by simulated annealing. At distances from the peptide that are beyond a few solvation shells, a continuum solvent model is employed. The simulations are performed using a mean field approach that enhances the efficiency of sampling peptide conformations. The computations predict a small number of conformations as plausible folded structures. All have a type VI turn conformation for the peptide backbone, similar to that found using NMR. However, some of the structures differ from the experimentally proposed ones in the packing of the proline ring with the aromatic residues. The second most populated structure has, in addition to a correctly folded backbone, the same hydrophobic packing as the conformation measured by NMR. Our simulations suggest a kinetic mechanism that consists of three separate stages. The time-scales associated with these stages are distinct and depend differently on temperature. Electrostatic interactions play an initial role in guiding the peptide chain to a roughly correct structure as measured by the end-to-end distance. At the same time or later the backbone torsions rearrange due to local tendency of the proline ring to form a turn: this step depends on solvation forces and is helped by loose hydrophobic interactions. In the final step, hydrophobic residues pack against each other. We also show the existence of an off the pathway intermediate, suggesting that even in the folding of a small peptide “misfolded” structures can form. The simulations clearly show that parallel folding paths are involved. Our findings suggest that the process of peptide folding shares many of the features expected for the significantly larger protein molecules.

BACKGROUND: Recent experimental and theoretical studies have shown that several small proteins reach the native state by a nucleation-collapse mechanism. Studies based on lattice models have been used to suggest that the critical nucleus is specific, leading to the notion that the transition state may be unique. On the other hand, results of studies using off-lattice models show that the critical nuclei should be viewed as fluctuating mobile structures, thus implying non-unique transition states. RESULTS: The microscopic underpinnings of the nucleation-collapse mechanism in protein folding are probed using minimal off-lattice models and Langevin dynamics. We consider a 46-mer continuum model which has a native beta-barrel-like structure. The fast-folding trajectories reach the native state by a nucleation-collapse process. An algorithm based on the self-organized neural nets is used to identify the critical nuclei for a large number of rapidly folding trajectories. This method, which reduces the determination of the critical nucleus to one of ‘pattern recognition’, unambiguously shows that the folding nucleus is not unique. The only common characteristics of the mobile critical nuclei are that they are small (containing on average 15-22 residues) and are largely composed of residues near the loop regions of the molecule. The structures of the transition states, corresponding to the critical nuclei, show the existence of spatially localized ordered regions that are largely made up of residues that are close to each other. These structures are stabilized by a few long-range contacts. The structures in the ensemble of transition states exhibit a rather diverse degree of similarity to the native conformation. CONCLUSIONS: The multiplicity of delocalized nucleation regions can explain the two-state folding by a nucleation-collapse mechanism for small single-domain proteins (such as chymotrypsin inhibitor 2) and their mutants. Because there are many distinct critical nuclei, we predict that the folding kinetics of fast-folding proteins will not be drastically changed even if some of the residues in a ‘typical’ nucleus are altered.

BACKGROUND: Recent experimental and theoretical studies have revealed that protein folding kinetics can be quite complex and diverse depending on various factors such as size of the protein sequence and external conditions. For example, some proteins fold apparently in a kinetically two-state manner, whereas others follow complex routes to the native state. We have set out to provide a theoretical basis for understanding the diverse behavior seen in the refolding kinetics of proteins in terms of properties that are intrinsic to the sequence. RESULTS: The folding kinetics of a number of sequences for off-lattice continuum models of proteins is studied using Langevin simulations at two different values of the friction coefficient. We show for these models that there is a remarkable correlation between folding time, tau F, and sigma = (T theta – TF)/T theta, where T theta and TF are the equilibrium collapse and folding transition temperatures, respectively. The microscopic dynamics reveals that several scenarios for the kinetics of refolding arise depending on the range of values of sigma. For relatively small sigma, the chain reaches the native conformation by a direct native conformation nucleation collapse (NCNC) mechanism without being trapped in any detectable intermediates. For moderate and large values of sigma, the kinetics is described by the kinetic partitioning mechanism, according to which a fraction of molecules phi (kinetic partition factor) reach the native conformation via the NCNC mechanism. The remaining fraction attains the native state by off-pathway processes that involve trapping in several misfolded structures. The rate-determining step in the off-pathway processes is the transition from the misfolded structures to the native state. The partition factor phi is also determined by sigma: the smaller the value of sigma, the larger is phi. The qualitative aspects of our results are found to be independent of the friction coefficient. The simulation results and theoretical arguments are used to obtain estimates for timescales for folding via the NCNC mechanism in small proteins, those with less than about 70 amino acid residues. CONCLUSIONS: We have shown that the various scenarios for folding of proteins, and possibly other biomolecules, can be classified solely in terms of sigma. Proteins with small values of sigma reach the native conformation via a nucleation collapse mechanism and their energy landscape is characterized by having one dominant native basin of attraction (NBA). On the other hand, proteins with large sigma get trapped in competing basins of attraction (CBAs) in which they adopt misfolded structures. Only a small fraction of molecules access the native state rapidly when sigma is large. For these sequences, the majority of the molecules approach the native state by a three-stage multipathway mechanism in which the rate-determining step involves a transition from one of the CBAs to the NBA.

We develop a heuristic model for chaperonin-facilitated protein folding, the iterative annealing mechanism, based on theoretical descriptions of “rugged” conformational free energy landscapes for protein folding, and on experimental evidence that (i) folding proceeds by a nucleation mechanism whereby correct and incorrect nucleation lead to fast and slow folding kinetics, respectively, and (ii) chaperonins optimize the rate and yield of protein folding by an active ATP-dependent process. The chaperonins GroEL and GroES catalyze the folding of ribulose bisphosphate carboxylase at a rate proportional to the GroEL concentration. Kinetically trapped folding-incompetent conformers of ribulose bisphosphate carboxylase are converted to the native state in a reaction involving multiple rounds of quantized ATP hydrolysis by GroEL. We propose that chaperonins optimize protein folding by an iterative annealing mechanism; they repeatedly bind kinetically trapped conformers, randomly disrupt their structure, and release them in less folded states, allowing substrate proteins multiple opportunities to find pathways leading to the most thermodynamically stable state. By this mechanism, chaperonins greatly expand the range of environmental conditions in which folding to the native state is possible. We suggest that the development of this device for optimizing protein folding was an early and significant evolutionary event.

We present a lattice Monte Carlo study to examine the effect of denaturants on the folding rates of simplified models of proteins. The two-dimensional model is made from a three-letter code mimicking the presence of hydrophobic, hydrophilic, and cysteine residues. We show that the rate of folding is maximum when the effective hydrophobic interaction epsilon H is approximately equal to the free energy gain epsilon S upon forming disulfide bonds. In the range 1 < or = epsilon H/ epsilon S < or = 3, multiple paths that connect several intermediates to the native state lead to fast folding. It is shown that at a fixed temperature and epsilon S the folding rate increases as epsilon H decreases. An approximate model is used to show that epsilon H should decrease as a function of the concentration of denaturants such as urea or guanidine hydrochloride. Our simulation results, in conjunction with this model, are used to show that increasing the concentration of denaturants can lead to an increase in folding rates. This occurs because denaturants can destabilize the intermediates without significantly altering the energy of the native conformation. Our findings are compared with experiments on the effects of denaturants on the refolding of bovine pancreatic trypsin inhibitor and ribonuclease T1. We also argue that the phenomenon of denaturant-enhanced folding of proteins should be general.

We use a three-dimensional lattice model of proteins to investigate systematically the global properties of the polypeptide chains that determine the folding to the native conformation starting from an ensemble of denatured conformations. In the coarse-grained description, the polypeptide chain is modeled as a heteropolymer consisting of N beads confined to the vertices of a simple cubic lattice. The interactions between the beads are taken from a random gaussian distribution of energies, with a mean value B0 < 0 that corresponds to the overall average hydrophobic interaction energy. We studied 56 sequences all with a unique ground state (native conformation) covering two values of N (15 and 27) and two values of B0. The smaller value of magnitude of B0 was chosen so that the average fraction of hydrophobic residues corresponds to that found in natural proteins. The higher value of magnitude of B0 was selected with the expectation that only the fully compact conformations would contribute to the thermodynamic behavior. For N = 15 the entire conformation space (compact as well as noncompact structures) can be exhaustively enumerated so that the thermodynamic properties can be exactly computed at all temperatures. The thermodynamic properties for the 27-mer chain were calculated using the slow cooling technique together with standard Monte Carlo simulations. The kinetics of approach to the native state for all the sequences was obtained using Monte Carlo simulations. For all sequences we find that there are two intrinsic characteristic temperatures, namely, T theta and Tf. At the temperature T theta the polypeptide chain makes a transition to a collapsed structure, while at Tf the chain undergoes a transition to the native conformation. We show that foldability of sequences can be characterized entirely in terms of these two temperatures. It is shown that fast folding sequences have small values of sigma = (T theta – Tf)/T theta whereas slow folders have larger values of sigma (the range of sigma is 0 < sigma < 1). The calculated values of the folding times correlate extremely well with sigma. An increase in sigma from 0.1 to 0.7 can result in an increase of 5-6 orders of magnitudes in folding times. In contrast, we demonstrate that there is no useful correlation between folding times and the energy gap between the native conformation and the first excited state at any N for any value of B0. In particular, in the parameter space of the model, many sequences with varying energy gaps, all with roughly the same folding time, can be easily engineered. Folding sequences in this model, therefore, can be classified based solely on the value of sigma. Fast folders have small values of sigma (typically less than about 0.1), moderate folders have values of sigma in the approximate range between 0.1 and 0.6, while for slow folders sigma exceeds 0.6. The precise boundary between these categories depends crucially on N and on the model. The range of sigma for fast folders decreases with the length of the chain. At temperatures close to Tf fast folders reach the native conformation via a native conformation nucleation collapse mechanism without forming any detectable intermediates, whereas only a fraction of molecule phi (T) reaches the native conformation by this process for moderate folders. The remaining fraction reaches the native state via three-stage multipathway process. For slow folders phi (T) is close to zero at all temperatures. The simultaneous requirement of native state stability and kinetic accessibility can be achieved at high enough temperatures for those sequences with small values of sigma. The utility of these results for de novo design of proteins is briefly discussed.

A simple continuum model of a de novo designed model of a four-helix bundle is presented. The thermodynamics and kinetics of the model are studied using Langevin simulations. We use a three-letter minimal off-lattice representation of a de novo designed four-helix bundle protein. The native state of the model, which can be thought of as an alpha-carbon representation of the peptide chain, is a caricature of the sequence designed by Ho and Degrado and shows several characteristics found in the naturally occurring four-helix bundles. These include the structural aspects and the relative stability of the native conformation. The model four-helix bundle shows two characteristic temperatures T theta and Tf. The former is the temperature above which the structure resembles that of the random coil. Below the first-order folding transition temperature Tf the chain adopts the native conformation corresponding to the four-helix bundle. It is shown that in order to obtain a unique native structure a proper free energy balance between secondary and tertiary interactions is needed. The thermal denaturation starting from the unique native conformation indicates that at least a three-state analysis is required. The intermediates in the equilibrium thermal denaturation are all found to be native-like. The kinetics of refolding starting from an ensemble of denatured states shows that the acquisition of the native conformation takes place via a kinetic partitioning mechanism. A fraction of molecules, phi, reaches the native state by a topology inducing nucleation collapse mechanism, while the remainder (1-phi) follows a complex three-stage multipathway process. We suggest, in accord with our earlier studies, that phi is essentially determined by the intrinsic temperature scales T theta and Tf. Our studies indicate that better design of proteins can be achieved by making T theta as close to Tf as possible. Experimental implications for de novo design of proteins are briefly discussed.

We propose a phenomenological theory that accounts for entropic effects due to loop formation to predict pathways in the kinetics of protein folding. The theory, the basis of which lies in multiple folding pathways and a three-stage kinetics, qualitatively reproduces most of the kinetic measurements in the refolding of bovine pancreatic trypsin inhibitor. The resulting pathways show that nonnative kinetic transients are involved in the productive routes leading to the formation of native intermediates. Our theory emphasizes the importance of the random origin of chain folding initiation structures in directing protein folding.

The role of disulfide bonds in directing protein folding is studied using lattice models. We find that the stability and the specificity of the disulfide bond interactions play quite different roles in the folding process: Under some conditions, the stability decreases the overall rate of folding; the specificity, however, by yielding a simpler connectivity of intermediates, always increases the rate of folding. This conclusion is intimately related to the selection mechanism entailed by entropic driving forces, such as the loop formation probability, and entropic barriers separating the native and the many native-like metastable states. The folding time is found to be a minimum for a certain range of the effective disulfide bond interaction. Examination of a model, which allows for the formation of disulfide bonded intermediates, suggests that folding proceeds via a three-stage multiple pathways kinetics. We show that there are pathways to the native state involving only native-like intermediates, as well as those that are mediated by nonnative intermediates. These findings are interpreted in terms of the appropriate energy landscape describing the barriers connecting low energy conformations. The consistency of our conclusions with several experimental studies is also discussed.

We present two methods to probe the energy landscape and motions of proteins in the context of molecular dynamics simulations of the helix-forming S-peptide of RNase A and the RNase A-3′-UMP enzyme-product complex. The first method uses the generalized ergodic measure to compute the rate of conformational space sampling. Using the dynamics of nonbonded forces as a means of probing the time scale for ergodicity to be obtained, we argue that even in a relatively short time (< 10 psec) several different conformational substrates are sampled. At longer times, barriers on the order of a few kcal/mol (1 cal = 4.184 J) are involved in the large-scale motion of proteins. We also present an approximate method for evaluating the distribution of barrier heights g(EB) using the instantaneous normal-mode spectra of a protein. For the S-peptide, we show that g(EB) is adequately represented by a Poisson distribution. By comparing with previous work on other systems, we suggest that the statistical characteristics of the energy landscape may be a “universal” feature of all proteins.

Monte Carlo simulations on a class of lattice models are used to probe the thermodynamics and kinetics of protein folding. We find two transition temperatures: one at T theta, when chains collapse from a coil to a compact phase, and the other at Tf (< T theta), when chains adopt a conformation corresponding to their native state. The kinetics are probed by several correlation functions and are interpreted in terms of the underlying energy landscape. The transition from the coil to the native state occurs in three distinct stages. The initial stage corresponds to a random collapse of the protein chain. At intermediate times tau c, during which much of the native structure is acquired, there are multiple pathways. For longer times tau r (> tau c) the decay is exponential, suggestive of a late transition state. The folding time scale (approximately tau r) varies greatly depending on the model. Implications of our results for in vitro folding of proteins are discussed.

The dynamic properties of the RNase A/3′-UMP enzyme/product complex and the S-peptide of RNase A have been investigated by molecular dynamics simulations using suitable generalization of ideas introduced to probe the energy landscape in structural glasses. We introduce two measures, namely, the kinetic energy fluctuation metric and the force metric, both of which are used to calculate the time needed for sampling the conformation space of the molecules. The calculation of the fluctuation metric requires a single trajectory whereas the force metric is computed using two independent trajectories. The vacuum MD simulations show that for both systems the time required for kinetic energy equipartitioning is surprisingly long even at high temperatures. We show that the force metric is a powerful means of probing the nature and relative importance of conformational substates which determine the dynamics at low temperatures. In particular the time dependence of the non-bonded force metric is used to demonstrate that at low temperatures the system is predominantly localized in a single cluster of conformational substates. The force metric is used to show that relaxation of long range (in sequence space) interactions must be mediated by a sequence of local dihedral angle transitions. We also argue that the time needed for compact structure formation is intimately related to the time needed for the relaxation of the dihedral angle degrees of freedom. The time for non-bonded interactions, which drive protein molecules to fold under appropriate conditions, to relax becomes extremely long as the temperature is lowered suggesting that the formation of maximally compact structure in proteins must be a very slow process.

We suggest, using dynamical simulations of a simple heteropolymer modelling the alpha-carbon sequence in a protein, that generically the folded states of globular proteins correspond to statistically well-defined metastable states. This hypothesis, called the metastability hypothesis, states that there are several free energy minima separated by barriers of various heights such that the folded conformations of a polypeptide chain in each of the minima have similar structural characteristics but have different energies from one another. The calculated structural characteristics, such as bond angle and dihedral angle distribution functions, are assumed to arise from only those configurations belonging to a given minimum. The validity of this hypothesis is illustrated by simulations of a continuum model of a heteropolymer whose low temperature state is a well-defined beta-barrel structure. The simulations were done using a molecular dynamics algorithm (referred to as the “noisy” molecular dynamics method) containing both friction and noise terms. It is shown that for this model there are several distinct metastable minima in which the structural features are similar. Several new methods of analyzing fluctuations in structures belonging to two distinct minima are introduced. The most notable one is a dynamic measure of compactness that can in principle provide the time required for maximal compactness to be achieved. The analysis shows that for a given metastable state in which the protein has a well-defined folded structure the transition to a state of higher compactness occurs very slowly, lending credence to the notion that the system encounters a late barrier in the process of folding to the most compact structure. The examination of the fluctuations in the structures near the unfolding—-folding transition temperature indicates that the transition state for the unfolding to folding process occurs closer to the folded state.

The possibility that several metastable minima exist in which the folded forms of a polypeptide chain have similar structural characteristics but different energies is suggested. The validity of this hypothesis is illustrated with the aid of simulation methods on a model protein that folds into a beta-barrel structure. Some implications of this hypothesis such as the existence of multiple pathways with intermediates for protein folding are discussed.